ElectroMagnetic Theory Made Easy.....

Antenna Theory - What is Polarization & What Are It's Different Types?

2012-01-01T18:33:00.002+05:30

Polarization is defined as the orientation of electric field as a function of direction. The polarization of the radio wave can be defined by direction in which the electric vector E is aligned during the passage of at least one full cycle. Also polarization can also be defined the physical orientation of the radiated electromagnetic waves in space.

The polarization are of three types. They are:

Elliptical polarization.
Circular polarization.
Linear polarization.

Linear Polarisation:

A linearly polarized wave is one in which the electric field remains in only one direction.For a linearly polarized wave,the axial ratio(AR) is infinity.

Elliptical Polarization:

The electric field vector rotates and form a ellipse called polarization ellipse.

The ratio of the major to the minor axes of the polarization ellipse is called the Axial Ratio (AR). AR is greater than 1 .

Circular Polarization:

The electric filed vector rotates and form a circle and this wave is called circularly polarized
wave. Axial Ratio (AR) is unity.

Antenna Theory - Effective Aperture & Directivity Of A Short Dipole Antenna.

2011-11-12T18:31:00.001+05:30

Consider a plane wave incident on a short dipole. The wave is assumed to be linearly polarized with electric field in the y direction. The current in the dipole is assumed constant and in the same phase over its entire length, and the terminating resistance is assumed equal to the dipole radiation resistance.

The effective aperture of this dipole is given by

A_e = 0.119 λ

The directivity is found to be

D = 4ΠA_e / sq (λ)

Frequently Asked Questions On Antenna Theory With Answers -20.

2011-11-12T18:18:00.001+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

96) What is Antenna matching?

When the antenna is receiving with a load resistance matched to the antenna radiation resistance, maximum power is transferred to the load and the power is also re-radiated from the dipole. This is called antenna matching.

97) What is a Short Dipole?

A short dipole is one in which the field is oscillating because of the oscillating voltage and current. It is called so, because the length of the dipole is short and the current is almost constant throughtout the entire length of the dipole.

98) How are fields created from short dipole / oscillating dipole?

The dipole has two equal charges of opposite sign oscillating up and down in a harmonic motion. The charges will move towards each other and electric filed lines were created. When the charges meet at the midpoint, the field lines cut each other and new field are created.This process is spontaneous and so more filed are created around the antenna. This is how radiations are obtained from a short dipole.

99) What are Antenna Field Zones?

The regions containing the radiations that are present around the antenna are called field zones. The fields around an antenna ay be divided into two principal regions.

Near field zone (Fresnel zone).
Far field zone (Fraunhofer zone).

100) What is self impedance and mutual impedance?

Self impedance of an antenna is defined as its input impedance with all other antennas are completely removed i.e away from it.

The presence of near by antenna no.2 induces a current in the antenna no.1 indicates that presence of antenna no.2 changes the impedance of the antenna no.1. This effect is called mutual coupling and
results in mutual impedance.

Antenna Theory - What is Antenna Aperture & What Are It's Types?

2011-11-12T17:50:00.001+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

Aperture represents the area of the antenna confining the effective radiations.

The various types of antenna apertures are:

Effective aperture.
Scattering aperture.
Loss aperture.
Collecting aperture.
Physical aperture.

Effective aperture (A_e):

It is the area over which the power is extrated from the incident wave and delivered to the load is called effective aperture.

Scattering aperture (A_s):

It is the ratio of the re radiated power to the power density of the incident wave.

Loss aperture (A_l):

It is the area of the antenna which dissipates power as heat.

Collecting aperture (A_c):

It is the addition of above three apertures.

Physical aperture (A_p):

This aperture is a measure of the physical size of the antenna.

The ratio of the effective aperture to the physical aperture is the aperture efficiency.
i.e.
Aperture efficiency = η_ap = A_e / A_p (dimensionless).

Frequently Asked Questions On Antenna Theory With Answers -19.

2011-11-12T17:23:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

91) Define Refractive index?

It is defined as n = c / V_p
where n = √ε_r

92)Define maximum Usable Frequency?

The maximum Frequency that can be reflected back for a given distance of transmission is called the maximum usable frequency (MUF) for that distance.

MUF = f_cr sec φ_i

93) Define skip distance?

The distance with in which a signal of given frequency fails to be reflected back is the skip distance for that frequency. The higher the frequency the greater the skip distance.

94) Define Optimum frequency?

Optimum frequency for transmitting between any two points is therefore selected as some frequency lying between about 50 and 85 percent of the predicted maximum usable frequency between those points.

95) What is wave impedance?

η = η_o / √ 1 - (f_c - f)

η = 377 / √ 1 - (f_c - f)

Frequently Asked Questions On Antenna Theory With Answers -18.

2011-11-12T17:03:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

86) What are the factors that affect the propagation of radio waves?

Curvature of earth.
Earth’ s magnetic field.
Frequency of the signal.
Plane earth reflection.

87) Define gyro frequency?

Frequency whose period is equal to the period of an electron in its orbit under the influence of the earths magnetic flux density B.

88) Define critical frequency?

For any layer, the highest frequency that will be reflected back for vertical incidence is
f_cr = 9 √N_max

89) Define Magneto-Ions Splitting?

The phenomenon of splitting the wave into two different components (ordinary and extra-ordinary) by the earths magnetic field is called Magneto-Ions Splitting.

90) Define LUHF?

The lowest useful HF for a given distance and transmitter power is defined as the lowest frequency that will give satisfactory reception for that distance and power.

It depends on:

The effective radiated power
Absorption character of ionosphere for the paths between transmitter and receiver.
The required field strength which in turn depends upon the radio noise at the receiving location and type of service involved.

Frequently Asked Questions On Antenna Theory With Answers -17.

2011-11-12T16:50:00.001+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

81) What is meant by diversity reception?

To minimize the fading and to avoid the multi path interference the technique used are diversity
reception. It is obtained by three ways:

Space diversity reception.
Frequency diversity reception.
Polarization diversity.

82) Define Space diversity Reception?

This method exploits the fact that signals received at different locations do not fade together. It requires antennas spaced at least 100λ apart are preferred and the antenna which high signal strength at the moment dominates.

83) Define frequency diversity Reception?

This method takes advantage of the fact that signals of slightly different frequencies do not fade synchronously. This fact is utilized to minimize fading in radio telegraph circuits.

84) Define polarization diversity reception?

It is used in normally in microwave links, and it is found that signal transmitted over the same path in two polarization have independent fading patterns. In broad band dish antenna system, Polarization diversity combined with frequency diversity reception achieve excellent results.

85) What is meant by Faraday's rotation?

Due to the earth’ s magnetic fields, the ionosheric medium becomes anisotropic and the incident plane wave entering the ionosphere will split into ordinary and extra ordinary waves/modes.

When these modes re-emerge from the ionosphere they recombine into a single plane wave again. Finally the plane of polarization will usually have changed, this phenomenon is known as Faraday's rotation.

Frequently Asked Questions On Antenna Theory With Answers -16.

2011-11-12T16:40:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

76) What is meant by Space Wave?

It is made up of direct wave and ground reflected wave. Also includes the portion of energy received as a result of diffraction around the earth surface and the reflection from the upper atmosphere.

77) What is meant by Surface Wave?

Wave that is guided along the earth’ s surface like an EM wave is guided by a transmission is called surface wave. Attenuation of this wave is directly affected by the constant of earth along which it travels.

78) What is meant by fading?

Fading is variation of signal strength occur on line of sight paths as a result of the atmospheric conditions. It can not be predicted properly.

79) What are the type of fading?

Two types:

Inverse fading.
Multi path fading.

80) What is inverse and multi path fading?

Inverse bending may transform line of sight path into an obstructed one.

Multi path fading is caused by interference between the direct and ground reflected waves as well as interference between two are more paths in the atmosphere.

Frequently Asked Questions On Antenna Theory With Answers -15.

2011-11-12T16:22:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

71) List the applications of helical antenna?

The applications of helical antenna are:

It became the workhouse of space communications for telephone, television and data, being employed both on satellites and at ground stations.
Many satellites including weather satellites, data relay satellites all have helical antennas.
It is on many other probes of planets and comets, including moon and mars, being used alone, in arrays or as feeds for parabolic reflectors, its circular polarization and high gain and simplicity making it effective for space application.

72) Define Sky wave?

Waves that arrive at the receiver after reflection in the ionosphere is called sky wave.

73) Define Tropospheric wave?

Waves that arrive at the receiver after reflection from the troposphere region is called Tropospheric wave (i.e. 10 Km from Earth surface).

74) Define Ground wave?

Waves propagated over other paths near the earth surface is called ground wave propagation.

75) What are the type of Ground wave?

Ground wave classified into two types.

Space wave.
Surface wave.

Frequently Asked Questions On Antenna Theory With Answers -14.

2011-11-12T16:13:00.001+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

66) What are Electrically large loop antennas?

Electrically Large loop antennas is one in which the overall length of the loop approaches the wavelength.

67) List out the uses of loop antenna?

Various uses of loop antenna are:

™ It is used as receiving antenna in portable radio and pagers.
™ It is used as probes for field measurements and as directional antennas for radio wave navigation.
™ It is used to estimate the direction of radio wave propagation.

68) What are the parameters to be considered for the design of an helical antenna?

The parameters to be considered for the design of an helical antenna are:

Bandwidth.
Gain.
Impedance.
Axial Ratio.

69) What are the types of radiation modes of operation for an helical antenna?

The two types of radiation modes of operation possible for an helical antenna are:

Normal mode of operation.
Axial mode of operation.

70) Which antenna will produce circularly polarized waves?

Helical antenna radiates circularly polarized wave

Frequently Asked Questions On Antenna Theory With Answers -13.

2011-11-12T16:02:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

61) What is a loop antenna?

A loop antenna is a radiating coil of any convenient cross-section of one or more turns carrying radio frequency current. It may assume any shape (e.g. rectangular, square, triangular and hexagonal).

62) Give an expression of radiation resistance of a small loop?

Radiation resistance of a small loop is given by
R_r = 31,200 ( A / λ² )²

63) How to increase the radiation resistance of a loop antenna?

The radiation resistance of a loop antenna can be increased by:

Increasing the number of turns.
Inserting a ferrite core of very high permeability with loop antenna’ s circumference which will rise the magnetic field intensity called ferrite loop.

64) What are the types of loop antennas?

Loop antennas are classified into:

Electrically Small (Circumference < λ / 10 )
Electrically Small (Dimension comparable to λ )

65) What are Electrically Small loop antennas?

Electrically Small loop antennas is one in which the overall length of the loop is less than one-tenth of the wavelength. Electrically Small loop antennas have small radiation resistances that are usually smaller than their loop resistances. They are very poor radiators and seldom employed for transmission in radio communication.

Frequently Asked Questions On Antenna Theory With Answers -12.

2011-11-12T15:45:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

56) Give the expression for the effective aperture of a short dipole?

The effective aperture of a short dipole is given by A_e = 0.119 λ².

57) What is a dipole antenna?

A dipole antenna may be defined as a symmetrical antenna in which the two ends are at equal potential relative to the midpoint.

58) What is a half wave dipole?

A half wave antenna is the fundamental radio antenna of metal rod or tubing or thin wire which has a physical length of half wavelength in free space at the frequency of operation.

59) Give the expression for the effective aperture of a Half wave Dipole?

The effective aperture of a half wave dipole is given by
A_e = 0.13 λ².

60) What is the radiation resistance of a half wave dipole?

The radiation resistance of a half wave dipole is given by
R_r = 73 ohm.

Frequently Asked Questions On Antenna Theory With Answers -11.

2011-11-12T15:23:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

51) What do you understand by retarded current?

Since,the short electric dipole is so short, the current which is flowing through the dipole is assumed to be constant throughout its length. The effect of this current is not felt instantaneous at a distance point only after an interval equal to the time required for the wave to propagate over the distance r is called the retardation time.

The retarded current [I]=I_o exp( jw^'( t - r / c ) )
Where w^'r / c is the phase retardation.

52)Define induction field?

The induction field will predominate at points close to the current element, where the distance from the center of the dipole to the particular point is less. This field is more effective in the vicinity of the current element only. It represents the energy stored in the magnetic field surrounding the current element or conductor. This field is also known as near field.

53) Define Radiation field?

The radiation field will be produced at a larger distance from the the current element, where the distance from the centre of the dipole to the particular point is very large. It is also called as distant field or far field.

54) At what distance from the dipole is the induction field equal to the radiation field?

As the distance from the current element or the short dipole increases, both induction and radiation fields emerge and start decreasing. However, a distance reaches from the conductor at which both the induction and radiation field becomes equal and the particular distance depends upon the wavelength. The two fields will thus have equal amplitude at that particular distance. This distance is given by r = 0.159 λ.

55) Define Radiation Resistance?

It is defined as the fictitious resistance which when inserted in series with the antenna will consume the same amount of power as it is actually radiated. The antenna appears to the transmission line as a resistive component and this is known as the radiation resistance.

Antenna Theory - Introduction To Short Dipole & How Radiation Are Created From It?

2011-11-10T16:47:00.000+05:30

A short dipole is one in which the field is oscillating because of the oscillating voltage and current. It is called so, because the length of the dipole is short and the current is almost constant throughtout the entire length of the dipole. It is also called as Hertzian Dipole which is a hypothetical antenna and is defined as a short isolated conductor carrying uniform alternating current.

The dipole has two equal charges of opposite sign oscillating up and down in a harmonic motion. The charges will move towards each other and electric filed lines were created. When the charges meet at the midpoint, the field lines cut each other and new field are created. This process is spontaneous and so more fields are created around the antenna. This is how radiations are obtained from a short dipole.

A short dipole that does have a uniform current is known as elemental dipole. Such a dipole will generally be considerably shorter than the tenth wave length maximum specified for a short
dipole. Elemental dipole is also called as elementary dipole, elementary doublet and hertzian dipole.

When the length of the short dipole is vanishingly small, then such a dipole is called a infinitesimal dipole. If dl be the infinitesimally small length and I be the current,then Idl is called as the current element.

A short dipole is initially in neutral condition and the moment a current starts to flow in one direction, one half of the dipole require an excess of charge and the other a deficit because a current is a flow of electrical charge. Then, there will be a voltage between the two halves of the dipole. When the current changes its direction this charge unbalance will cause oscillations. Hence an oscillating current will result in an oscillating voltage. Since, in such dipole, electric charge oscillates, it may be called as Oscilllating electric dipole.

Frequently Asked Questions On Antenna Theory With Answers -10.

2011-11-10T16:32:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

46) What is a binomial array?

It is an array in which the amplitudes of the antenna elements in the array are arranged according to the coefficients of the binomial series.

47) What are the advantages of binomial array?

Advantage:

No minor lobes.

Disadvantages:

Increased beam width.
Maintaining the large ratio of current amplitude in large arrays is difficult.

48) What is the difference between isotropic and non-isotropic source?

Isotropic source radiates energy in all directions but non-isotropic source radiates energy only in some desired directions.
Isotropic source is not physically realizable but non-isotropic source is physically realizable.

49) Define Side Lobe Ratio?

Side Lobe Ratio is defined as the ratio of power density in the principal or main lobe to the power
density of the longest minor lobe.

50) List the arrays used for array tapering?

Binomial Array:Tapering follows the coefficient of binomial series.
Dolph Tchebycheff Array: Tapering follows the coefficient of Tchebycheff polynomial.

Frequently Asked Questions On Antenna Theory With Answers - 9.

2011-11-10T16:23:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

41) What is meant by similar Point sources?

Whenever the variation of the amplitude and the phase of the field with respect to the absolute angle for any two sources are same then they are called similar point sources.

The maximum amplitudes of the individual sources may be unequal.

42) What is meant by identical Point sources?

Similar point sources with equal maximum amplitudes are called identical point sources.

43) What is the principle of the pattern multiplication?

The total field pattern of an array of non isotropic but similar sources is the product of the

Individual source pattern and
The array pattern of isotropic point sources each located at the phase center of the individual
source having the same amplitude and phase.

While the total phase pattern is the sum of the phase patterns of the individual source pattern and array pattern.

44) What is the advantage of pattern multiplication?

Useful tool in designing antenna.
It approximates the pattern of a complicated array without making lengthy computations.

45) What is tapering of arrays?

Tapering of array is a technique used for reduction of unwanted side lobes. The amplitude of currents in the linear array source is non-uniform; hence the central source radiates more energy than the ends. Tapering is done from center to end.

Frequently Asked Questions On Antenna Theory With Answers - 8.

2011-11-10T16:12:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

36) Define beam width of major lobe?

It is defined the angle between the first nulls (or) it is defined as twice the angle between the first null and the major lobe maximum direction.

37) List out the expression of beam width for broad side array and end fire array?

For broad side array the expression for beam width between the first nulls is given by,

BWFN = ( ( + / -) 2 λ / n d )

For End fire array the expression for beam width between the first nulls is given by,

BWFN = ( ( + / -) 2 ( 2 λ / n d ) )^1/2

38) Differentiate broad side and End fire array?

In Broad side array antennas are fed in phase δ = 0, where as in end fire arrays the antenna elements are fed out of phase i.e. δ = - β d.
In broad side array the maximum radiation is perpendicular to the direction of array axis, where as in case of end fire array the maximum radiation is directed along the array axis.

39) What is the need for the Binomial array?

The need for a binomial array is

In uniform linear array as the array length is increased to increase the directivity, the secondary lobes also occurs.
For certain applications, it is highly desirable that secondary lobes should be eliminated completely or reduced to minimum desirable level compared to main lobes.

40) Define power pattern?

Graphical representation of the radial component of the pointing vector Sr at a constant radius as
a function of angle is called power density pattern or power pattern.

Frequently Asked Questions On Antenna Theory With Answers -7.

2011-11-10T15:59:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

31) Define End fire array?

End fire array is defined as an arrangement in which the principal direction of radiation is coincides with the array axis.

For end fire array δ = - β d
where β = 2Π / λ and d = distance between the elements.

32) What is collinear array?

In this array the antenna elements are arranged coaxially by mounting the elements end to end in straight line or stacking them one over the other with radiation pattern circular symmetry.
Eg. Omnidirectional antenna.

33) What is Parasitic array?

In this array the elements are fed parasitically to reduce the problem of feed line. The power is given to one element from that other elements get by electro magnetic coupling.
Eg. Yagi uda antenna.

34) What is the condition on phase for the end fire array with increased directivity?

When δ = - β d, produces maximum field in the direction φ = 0 but dies not give the maximum directivity. It has been shown by Hansen and woodyard that a large directivity is obtained by increasing the phase change between the sources so that δ = - (βd + π / n ).

This condition will be referred to as the condition for increased directivity.

35) Define array factor?

The normalized value of the total field is given by,
E = ( 1 / n) ( sin (n Ψ / 2) / sin ( Ψ / 2) )

The field is given by the expression E will be referred to as array factor.

Frequently Asked Questions On Antenna Theory With Answers - 6.

2011-11-10T15:35:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

26) What is point source?

It is the waves originate at a fictitious volume less emitter source at the center ‘O’ of the observation circle.

27) What is meant by array?

An antenna is a system of similar antennas oriented similarly to get greater directivity in a desired direction.

28) What is meant by uniform linear array?

An array is linear when the elements of the array are spaced equally along the straight line. If the elements are fed with currents of equal magnitude and having a uniform progressive phase shift along the line, then it is called uniform linear array .

29) What are the types of array?

Broad side array.
End fire array
Collinear array.
Parasitic array.

30) What is Broad side array?

Broad side array is defined as an arrangement in which the principal direction of radiation is perpendicular to the array axis and also the plane containing the array element.

Frequently Asked Questions On Antenna Theory With Answers - 5.

2011-11-10T14:52:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

21) What is meant by cross field?

Normally the electric field E is perpendicular to the direction of wave propagation. In some situation the electric field E is parallel to the wave propagation that condition is called Cross field.

22) Define axial ratio?

The ratio of the major to the minor axes of the polarization ellipse is called the Axial Ratio (AR).

23) What is meant by Beam Area?

The beam area or beam solid angle or WA of an antenna is given by the normalized power pattern over a sphere.

WA = ò ò4p Pn ( q,f ) dW
Where dW = Sin q dq.df

24) What is duality of antenna?

It is defined as an antenna is a circuit device with a resistance and temperature on the one hand and the space device on the other with radiation patterns, beam angle ,directivity gain and aperture.

25) State Poynting theorem?

It states that the vector product of electric field intensity vector E and the magnetic filed intensity vector H at any point is a measure of the rate of energy flow per unit area at that point. The direction of power flow is perpendicular to both the electric field and magnetic field components.

Frequently Asked Questions On Antenna Theory With Answers - 4.

2011-11-10T14:41:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

16) What is meant by reciprocity Theorem?

If an e.m.f is applied to the terminals of an antenna no.1 and the current measured at the terminals of the another antenna no.2, then an equal current both in amplitude and phase will be obtained at the terminal of the antenna no.1 if the same emf is applied to the terminals of antenna no.2.

17) What is meant by isotropic radiator?

A isotropic radiator is a fictitious radiator and is defined as a radiator which radiates fields uniformly in all directions. It is also called as isotropic source or omni directional radiator or simply unipole.

18) Define gain?

The ratio of maximum radiation intensity in given direction to the maximum radiation intensity from a reference antenna produced in the same direction with same input power. i.e Maximum radiation intensity from test antenna (G)= Maximum radiation intensity from the reference antenna with same input power.

19) Define self impedance?

Self impedance of an antenna is defined as its input impedance with all other antennas are completely removed i.e away from it.

20) Define mutual impedance?

The presence of near by antenna no.2 induces a current in the antenna no.1 indicates that presence of antenna no.2 changes the impedance of the antenna no.1. This effect is called mutual coupling and results in mutual impedance.

Frequently Asked Questions On Antenna Theory With Answers -3.

2011-11-10T14:33:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

11) What is meant by Polarization?

The polarization of the radio wave can be defined by direction in which the electric vector E is aligned during the passage of atleast one full cycle.Also polarization can also be defined the physical orientation of the radiated electromagnetic waves in space. The polarization are three types. They are Elliptical polarization ,circular polarization and linear polarization.

12) What is meant by front to back ratio?

It is defined as the ratio of the power radiated in desired direction to the power radiated in the opposite direction. i.eFBR = Power radiated in desired direction / power radiated in the opposite direction.

13) Define antenna efficiency?

The efficiency of an antenna is defined as the ratio of power radiated to the total input power supplied to the antenna. Antenna efficiency = Power radiated / Total input Power

14) What is radiation resistance ?

The antenna is a radiating device in which power is radiated into space in the form of electromagnetic wave. W’ = I2 R Rr = W’/ I2 Where Rr is a fictitious resistance called called as radiation resistance.

15) What is meant by antenna beam width?

Antenna beamwidth is a measure of directivity of an antenna. Antenna beam width is an angular width in degrees, measured on the radiation pattern (major lobe) between points where the radiated power has fallen to half its maximum value. This is called as “beam width” between half power points or half power beam width.(HPBW).

Frequently Asked Questions On Antenna Theory With Answers -2.

2011-11-10T14:26:00.000+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

6) What are the different types of aperture?

Effective aperture.
Scattering aperture.
Loss aperture.
Collecting aperture.
Physical aperture.

7) Define different types of aperture?

Effective aperture (Ae):

It is the area over which the power is extracted from the incident wave and delivered to the load is called effective aperture.

Scattering aperture (As):

It is the ratio of the reradiated power to the power density of the incident wave.

Loss aperture (Ae):

It is the area of the antenna which dissipates power as heat.

Collecting aperture (Ae):

It is the addition of above three apertures.

Physical aperture (Ap):

This aperture is a measure of the physical size of the antenna.

8) Define Aperture efficiency?

The ratio of the effective aperture to the physical aperture is the aperture efficiency. i.e Aperture efficiency = hap = Ae / Ap (dimensionless).

9) What is meant by effective height?

The effective height h of an antenna is the parameter related to the aperture. It may be defined as the ratio of the induced voltage to the incident field.i.e H= V / E.

10) What are the field zones?

The fields around an antenna ay be divided into two principal regions.

Near field zone (Fresnel zone)
Far field zone (Fraunhofer zone)

Frequently Asked Questions On Antenna Theory With Answers -1.

2011-11-10T14:20:00.001+05:30

This page contains basic questions and short notes on some of the keywords on antenna theory:

1) Define an antenna?

Antenna is a transition device or a transducer between a guided wave and a free space wave or vice-versa. Antenna is also said to be an impedance transforming device.

2) What is meant by radiation pattern?

Radiation pattern is the relative distribution of radiated power as a function of distance in space .It is a graph which shows the variation in actual field strength of the EM wave at all points which are at equal distance from the antenna. The energy radiated in a particular direction by an antenna is measured in terms of FIELD STRENGTH. (E Volts/m)

3) Define Radiation intensity?

The power radiated from an antenna per unit solid angle is called the radiation intensity U (watts per steradian or per square degree). The radiation intensity is independent of distance.

4) Define Beam efficiency?

The total beam area (WA) consists of the main beam area (WM) plus the minor lobe area (Wm).
Thus WA = WM+ Wm.

The ratio of the main beam area to the total beam area is called beam efficiency.
Beam efficiency (SM) = WM / WA.

5) Define Directivity?

The directivity of an antenna is equal to the ratio of the maximum power density P (q,f)max to its average value over a sphere as observed in the far field of an antenna.

D = P (q,f)max / P(q,f)av. Directivity from Pattern.

D = 4p / WA. Directivity from beam area (WA).

Field Theory - Conductor ( Isolated and Non-Isolated) - Under the Influence Of Electric Field...

2010-02-18T00:34:00.006+05:30

- A conductor (such as Copper, Aluminum, gold) contains electrons that are not bounded to the atom and are free to move within the material.

- When an isolated conductor is placed in an external electric field E_e, the negative charges (electrons) starts moving in the opposite direction to that of electric field.

- Soon the electrons get accumulated on the surface of one side of the conductor, while the surface on the other side gets depleted of electrons and hence acquire a positive charge (+ve).

- These separated negative (-ve) and positive charges (+ve) on the opposite sides of the conductor produces an internal induced electric field E_i which opposes the external field E_e inside the conductor.

- If the conductor is a perfect conductor or a good conductor the internal electric field E_i will cancel the external electric field E_e. Hence we can therefore say that

“A perfect conductor cannot contain an electrostatic field inside it.”

- There is a net field of zero magnitude inside the conductor. Hence if E = 0, then the electric flux through any arbitrary closed surface inside the conductor is also equal to zero. This immediately implies that the charge density ρ_v inside the conductor is equal to zero everywhere (Gauss's law).

- If any excess charge is placed within the volume of the conductor, the repulsive forces pushes them far apart. Since the farthest is the surface itself, all the excess charge resides entirely on the surface. Hence it’s clear that the net charge on an isolated conductor exits only on the surface.

- The electrostatic field at the conductor’s surface is proportional to the surface charge and does not depend on the charge carriers inside the conductor.

- It’s safe to be inside a car / automobile during a thunderstorm. Why?

Ans: Yes, because the metal body of the car carries the excess charge on its external surface. Also we know inside a closed conducting surface electric field is zero. Hence the metal body acts like a screen or shield for the occupants.

PROPERTIES OF AN ISOLATED CONDUCTOR (IN STATIC SITUATION):

- Electric field is zero everywhere within the conductor and also inside a closed conducting surface.

- Any excess charge resides only on the surface of the conductor.

- Electric field just outside the charged conductor is perpendicular to the conductor’s surface.

- On an irregular shaped conductor, the charges tend to accumulate at locations where the radius of curvature of the surface is smallest i.e. at sharp points. Hence we can also say that the field is strongest on the pointy parts of a conductor.

- There is no potential difference between any two points in the conductor i.e. a conductor is an equipotential body.

NON - ISOLATED CONDUCTOR

OR

CONDUCTOR IN A NON-ELECTROSTATIC EQUILIBRIUM:

- Consider a conductor having a uniform cross section S, length l and whose ends are maintained at a potential difference V as shown in the figure:

- When a potential difference of V is applied across the conductor, the electric field inside a conductor is not zero. Hence there is no static equilibrium inside a conductor.

- A conductor is said to be in electrostatic equilibrium only if no electric field exist inside a conductor.

- Since E ≠ 0, the free charges in the conductor start moving, thus producing conduction current.

- As the electrons moves from one end to another, they experience a damping force called Resistance.

- The electric field E applied is uniform (since V is constant) and is related to electric potential V as:

E = V / l

- Since the conductor has a uniform cross section, the current density is given as:

J = I / S

- Also conduction current density (J) is given as:

J = σ E

- Hence substituting the value of J in the above equation, we have

I = σ E S

Equating the expressions for E

E = V / l = I / σ S

→ V = (l / σ S) I

→ V = I R

- Hence Resistance of a conductor is given as:

R = l / σ S = ρ l / S

Where σ is the conductivity and ρ is the resistivity of the conductor.

- For a conductor of non-uniform cross section , the resistance is given as:

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Field Theory - Conduction And Convection Current Density....

2010-02-17T18:00:00.004+05:30

CONVECTION CURRENT DENSITY:

- Convection current occurs in insulators or dielectrics such as liquid, vacuum and rarified gas.

- Convection current results from motion of electrons or ions in an insulating medium.

- Since convection current doesn’t involve conductors, hence it does not satisfy ohm’s law.

- Consider a filament where there is a flow of charge ρ_v at a velocity u = u_y a_y.

- Hence the current is given as:

Where u_y is the velocity of the moving electron or ion and ρ_v is the free volume charge density.

- Hence the convection current density in general is given as:

J = ρ_v u

CONDUCTION CURRENT DENSITY:

- Conduction current occurs in conductors where there are a large number of free electrons.

- Conduction current occurs due to the drift motion of electrons (charge carriers).

- Conduction current obeys ohm’s law.

- When an external electric field is applied to a metallic conductor, conduction current occurs due to the drift of electrons.

- The charge inside the conductor experiences a force due to the electric field and hence should accelerate but due to continuous collision with atomic lattice, their velocity is reduced.

- The net effect is that the electrons moves or drifts with an average velocity called the drift velocity (υ_d) which is proportional to the applied electric field (E).

- Hence according to Newton’s law, if an electron with a mass m is moving in an electric field E with an average drift velocity υ_d, the the average change in momentum of the free electron must be equal to the applied force (F = - e E).

- The drift velocity per unit applied electric field is called the mobility of electrons (μ_e).

υ_d = - μ_e E

where μ_e is defined as:

- Consider a conducting wire in which charges subjected to an electric field are moving with drift velocity υ_d.

- Say there are N_e free electrons per cubic meter of conductor, then the free volume charge density(ρ_v) within the wire is

ρ_v = - e N_e

- The charge ΔQ is given as:

ΔQ = ρ_v ΔV = - e N_e ΔS Δl = - e N_e ΔS υ_d Δt

- The incremental current is thus given as:

- The conduction current density is thus defined as:

where σ is the conductivity of the material.

- The above equation is known as the Ohm’s law in point form and is valid at every point in space.

- In a semiconductor, current flow is due to the movement of both electrons and holes, hence conductivity is given as:

σ = ( N_e μ_e + N_h μ_h )e

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- Conductor Under The Influence Of An Applied Electric Field (E).

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Field Theory - Properties Of Materials And Steady Electric Current...

2010-02-16T02:42:00.003+05:30

- Electric field can not only exist in free space and vacuum but also in any material medium.

- When an electric field is applied to the material, the material will modify the electric field either by strengthening it or weakening it, depending on what kind of material it is.

- Materials are classified into 3 groups based on conductivity / electrical property:

          - Conductors (Metals like Copper, Aluminum, etc.) have high conductivity (σ >> 1).
          - Insulators / Dielectric (Vacuum, Glass, Rubber, etc.) have low conductivity (σ << 1).
          - Semiconductors (Silicon, Germanium, etc.) have intermediate conductivity.

- Conductivity (σ) is a measure of the ability of the material to conduct electricity. It is the reciprocal of resistivity (ρ).

- Units of conductivity are Siemens/meter and mho.

- The basic difference between a conductor and an insulator lies in the amount of free electrons available for conduction of current.

- Conductors have a large amount of free electrons where as insulators have only a few number of electrons for conduction of current.

- Most of the conductors obey ohm’s law. Such conductors are also called ohmic conductors.

- Due to the movement of free charges, several types of electric current can be caused. The different types of electric current are:

          - Conduction Current.
          - Convection Current.
          - Displacement Current.

- Electric current (I) defines the rate at which the net charge passes through a wire of cross sectional surface area S.

Mathematically,
If a net charge ΔQ moves across surface S in some small amount of time Δt, electric current(I) is defined as:

- How fast or how speed the charges will move depends on the nature of the material medium.

- Current density (J) is defined as current ΔI flowing through surface ΔS.

- Imagine surface area ΔS inside a conductor at right angles to the flow of current. As the area approaches zero, the current density at a point is defined as:

The above equation is applicable only when current density (J) is normal to the surface.

- In case if current density(J) is not perpendicular to the surface, consider a small area ds of the conductor at an angle θ to the flow of current as shown:

In this case current flowing through the area is given as:

dI = J dS cosθ = J . dS

I = ∫_S (J . dS)

where angle θ is the angle between the normal to the area and direction of the current.

From the above equation it’s clear that electric current is a scalar quantity.

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- Conductor Under The Influence Of An Applied Electric Field (E).

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Field Theory - Numericals / Solved Examples - Electrostatic Energy and Energy Density...

2010-02-10T13:00:00.002+05:30

Q.1 Point charges Q₁ = 1 nC, Q₂ = -2 nC and Q₃ = 3 nC and are positioned one at a time and in that order at (0, 0, 0), (1, 0, 0) and (0, 0, -1) respectively. Calculate the energy in the system after each charge is positioned?
Ans:

Initially the system is assumed to be charge free.

The energy required to bring Q₁ into the system is 0 J.

After Q₁: Energy in the system = 0 J

The energy required to bring Q₂ into the system is

Hence, After Q₂: Energy in the system = -18 nJ

The energy required to bring Q₃ into the system is:

W₃ = Q₃( V₃₁ + V₃₂) + Q₂ V₂₁

Q2. Determine the work necessary to transfer charges Q₁ = 1 mC and Q₂ = -2 mC from infinity to points (-2, 6, 1) and (3, -4, 0) respectively.
Ans:

No work is done in transferring the first charge Q₁. However work done to transfer the point charge Q₂ is given as:

Q3. A point charge Q is placed at the origin. Calculate the energy stored in region r > a?
Ans:

Work done in assembling a volume charge distribution in terms of electric field and flux density is given as:

Electric field intensity due to a point charge Q placed at origin is given as:

Hence energy stored in a region r > a is given as:

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Field Theory - Energy Density In Electrostatic Field / Work Done To Assemble Charges....

2010-02-09T16:12:00.002+05:30

- In case, if we wish to assemble a number of charges in an empty system, work is required to do so.

- Also electrostatic energy is said to be stored in such a collection.

- Let us build up a system in which we position three point charges Q₁, Q₂ and Q₃ at position r₁, r₂ and r₃ respectively in an initially empty system.

- Consider a point charge Q₁ transferred from infinity to position r₁ in the system. It takes no work to bring the first charge from infinity since there is no electric field to fight against (as the system is empty i.e. charge free).

Hence, W₁ = 0 J

- Now bring in another point charge Q₂ from infinity to position r₂ in the system. In this case we have to do work against the electric field generated by the first charge Q₁.

Hence, W₂ = Q₂ V₂₁

where V₂₁ is the electrostatic potential at point r₂ due to Q₁.

- Work done W₂ is also given as:

- Now bring in another point charge Q₃ from infinity to position r₃ in the system. In this case we have to do work against the electric field generated by Q₁ and Q₂.

Hence, W₃ = Q₃ V₃₁ + Q₃ V₃₂ = Q₃ ( V₃₁ + V₃₂)

where V₃₁ and V₃₂ are electrostatic potential at point r₃ due to Q₁ and Q₂ respectively.

- The work done is simply the sum of the work done against the electric field generated by point charge Q₁ and Q₂ taken in isolation:

- Thus the total work done in assembling the three charges is given as:

W_E = W₁ + W₂ + W₃

= 0 + Q₂ V₂₁ + Q₃( V₃₁ + V₃₂ )

- Also total work done ( W_E ) is given as:

- If the charges were positioned in reverse order, then the total work done in assembling them is given as:

W_E = W₃ + W₂+ W₁

= 0 + Q₂V₂₃ + Q₃( V₁₂+ V₁₃)

where V₂₃ is the electrostatic potential at point r₂ due to Q₃ and V₁₂ and V₁₃ are electrostatic potential at point r₁ due to Q₂ and Q₃ respectively.

- Adding the above two equations we have,

2W_E = Q₁ ( V₁₂ + V₁₃) + Q₂ ( V₂₁ + V₂₃) + Q₃ ( V₃₁ + V₃₂)

= Q₁ V₁ + Q₂ V₂ + Q₃ V₃

Hence

W_E =1 / 2 [Q₁V₁ + Q₂V₂ + Q₃V₃]

where V₁, V₂ and V₃ are total potentials at position r₁, r₂ and r₃ respectively.

- The result can be generalized for N point charges as:

- The above equation has three interpretation:

a) This equation represents the potential energy of the system.

          b) This is the work done in bringing the static charges from infinity and assembling them in the required system.

          c) This is the kinetic energy which would be released if the system gets dissolved i.e. the charges returns back to infinity.

- In place of point charge, if the system has continuous charge distribution ( line, surface or volume charge), then the total work done in assembling them is given as:

- Since ρ_v = ∇ . D and E = - ∇ V,

Substituting the values in the above equation, work done in assembling a volume charge distribution in terms of electric field and flux density is given as:

- The above equation tells us that the potential energy of a continuous charge distribution is stored in an electric field.

- The electrostatic energy density w_E is defined as:

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Field Theory - Numericals/Solved Examples - Electric Potential, Potential Gradient and Electric dipole...

2010-02-08T17:50:00.004+05:30

Q.1 Determine the electric field due to the following potential:
a) V = x² + 2y² + 4z²
Ans:

= - (2x a_x + 4y a_y + 8z a_z)
= -2x a_x - 4y a_y - 8z a_z V/m

b) V = ρ² (z + 1) sinφ

= - [ 2ρ (z + 1)sinφ a_ρ + ρ(z + 1) cosφ a_φ + ρ² sinφ a_z ]
= - 2ρ (z + 1)sinφ a_ρ - ρ(z + 1) cosφ a_φ - ρ² sinφ a_z

Q.2 A point charge of 5 nC is located at the origin. If V = 2v at (0, 6, -8), find
a) The potential at A (-3, 2, 6)
b) The potential at B (1, 5, 7)
c) The potential difference V_AB
Ans:
If V (0, 6, -8) = 2 V
In this case
x = 0; y = 6; z = -8

Using the scalar relationship between Cartesian and spherical system, we have

r = (x² + y² +z²)^1/2 = (100)^1/2 =10

Hence

Electric potential at point r due to a point charge Q located at origin

→ C = - 2.5

a) Electric potential at point r due to a point charge Q located at a point (-3, 2, 6) is given as:

= 3. 929 V

b) Electric potential at point r due to a point charge Q located at a point (1, 5, 7) is given as:

= 2.696 V

c) The potential difference V_AB is given as:

V_AB = V_B – V_A = 2.696 – 3.929 = - 1.233 V

Q.3 If point charge 3 μC is located at the origin. Also there are two more charges -4 μC and 5 μC are located at (2, -1, 3) and (0, 4, -2) respectively. Find potential at (-1, 5, 2) ? Assume zero potential at infinity.
Ans:
For N point charges Q₁, Q₂ ….Q_n located at points with position vectors r₁, r₂, r₃…..r_n, the electric potential at point r is given as:

At V ( ∞ ) = 0, C = 0

| r – r₁ | = | (-1, 5, 2) – (2, -1, 3) | = (46)^1/2
| r – r₂ | = | (-1, 5, 2) – (0, 4, -2) | = (18)^1/2
| r – r₃ | = | (-1, 5, 2) – (0, 0, 0) | = (30)^1/2

Hence electric potential is given as:

= 10.3 kV

Q.4 An electric dipole of 100 a_z pC.m is located at the origin. Find V and E at points
a) (0, 0, 10)
b) (1, π/3, π/2)
Ans:
a) At point (0, 0, 10)
Cartesian → Spherical
(x, y, z) → (r, θ, φ)

(0, 0, 10) → (10, 0^o, 0^o)

Electric potential (V) due to a electric dipole centered at origin and aligned with the z axis is written as:

Electric field intensity (E) is the negative gradient of Electric Potential (V).

b)At point (1, π/3, π/2)

Electric potential (V) due to a electric dipole centered at origin and aligned with the z axis is written as:

Electric field intensity (E) is the negative gradient of Electric Potential (V).

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Field Theory - Electric Dipole....

2010-01-18T00:49:00.003+05:30

- An electric dipole consists of two point charges of equal magnitude but of opposite sign and separated by a small distance.

- Consider an electric dipole centered at origin and placed in z – axis as shown in the figure:

- The potential (V) at point P is given as:

- If the distance between the charges (d) is very small as compared to the distance of the point P from the origin i.e.

If r >> d,

r₂ – r₁ ≅ d cosθ ; r₁ ≅ r₂ = r ; r₁r₂ ≅ r²

Substituting the values in the above equation, the potential at point P becomes:

- Electric field intensity (E) is the negative gradient of Electric Potential (V).
Hence,

- The expressions for electric potential (V) and field intensity (E) above are only valid for a dipole centered at the origin and aligned with the z-axis.

- To determine the fields produced by any arbitrary location and alignment, we first need to define a new quantity p, called the Dipole Moment.

p = Q d

- Since the distance d is a vector quantity, the dipole moment p is also a vector quantity.

- Dipole moment p is a measure of the strength of the dipole and indicates its direction.

- Vector d is a directed distance that extends from negative charge (- Q) to positive charge (+ Q). This directed distance vector d thus describes the distance between the dipole charges, as well as the orientation of the charges.

Therefore

d = | d | a_d

Where | d | is the distance between the charges and a_d defines the orientation or direction of the dipole.

- Say a dipole is aligned along z – axis, then directed distance d is given as:

d = | d | a_z

From the above diagram it’s clear that:

a_z . a_r = cosθ

Hence the expression can be written as:

Q d cosθ = Q | d | a_z . a_r = Q d. a_r = p . a_r

- Hence electric potential (V) due to a electric dipole centered at origin and aligned with the z axis is rewritten as:

- The above expression no doubt is applicable for all and any dipole moments p, but is valid for dipoles centered at origin.

- Electric potential (V) at point P with a position vector r due to a dipole centered at a point with position vector r₁ is given as:

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Field Theory - Electric Potential (V) Due To A Circular Disc...

2010-01-15T20:28:00.004+05:30

- Consider a circular disc of radius ‘a’ which carries a uniform surface charge density ρ_s , C /m².

- Say the disk lies on x - y plane (or z = 0 plane) with its axis along the z axis as shown in the figure.

- We need to find out electric potential (V) due to a circular disk at a point P (0, 0, h) on the z axis (z > 0).

- Electric potential (V) at a point due to any surface charge (ρ_s) is given as:

- In this case,

ds = ρ dρ dφ

(Since it’s a disc, the varying terms are radius ρ and angle φ)

R = (ρ² + h²)^1/2

- Hence electric potential (V) is given as:

- On solving further the equation becomes

- As a → 0, electric potential (V) also tends to zero i.e. V → 0.

- Hence the electric potential at point (0, 0, h) is given as:

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- Electric Dipole....

- Numericals/Solved Examples - Electric Potential, Potential Gradient and Electric dipole...

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Field Theory - Relationship Between Electric Field Intensity(E) and Electric Potential(V)...

2010-01-14T23:57:00.004+05:30

- The work done per unit charge in moving a test charge from point A to point B is the electrostatic potential difference between the two points(V_AB).

V_AB = V_B - V_A

Similarly,

V_BA = V_A – V_B

- Hence it’s clear that potential difference is independent of the path taken.
Therefore

V_AB = - V_BA

V_AB + V_BA = 0

- ∫_A^B (E . dl) + [ - ∫_B^A (E . dl) ] = 0

- The above equation shows that the line integral of Electric field intensity (E) along a closed path is equal to zero.

- In simple words,

“No work is done in moving a charge along a closed path in an electrostatic field”.

- Applying Stokes’ Theorem to the above Equation, we have:

- If the Curl of any vector field is equal to zero, then such a vector field is called an Irrotational or Conservative Field.

- Hence an electrostatic field is also called a conservative field.

- The above equation is called the second Maxwell’s Equation of Electrostatics.

- Since Electric potential is a scalar quantity, hence dV (as a function of x, y and z variables) can be written as:

- Hence the Electric field intensity (E) is the negative gradient of Electric potential (V).

- The negative sign shows that E is directed from higher to lower values of V i.e. E is opposite to the direction in which V increases.

EQUIPOTENTIAL SURFACE:

- An equipotential surface refers to a surface where the potential is constant.

- The intersection of an equipotential surface and a plane results into a path called an equipotential line.

- No work is done in moving a charge from one point to the other along an equipotential line or surface i.e. V_A – V_B = 0

Hence,

From the above equation, it’s clear that the electric flux lines and the equipotential surface and normal to each other.

- Because the electric field is the negative gradient of electric potential, the electric field lines are everywhere normal to the equipotential surface and points in the direction of decreasing potential.

- The equipotential lines for a positive point charge. The solid lines show the flux lines or electric lines of force.

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Field Theory - Scalar Electric Potential / Electrostatic Potential (V)...

2010-01-13T22:49:00.006+05:30

- If a charge is placed in the vicinity of another charge (or in the field of another charge), it experiences a force.

- If a field being acted on by a force is moved from one point to another, then work is either said to be done on the system or by the system.

- Say a point charge Q is moved from point A to point B in an electric field E, then the work done in moving the point charge is given as:

W_A→B = - ∫_A^B (F . dl) = - Q ∫_A^B(E . dl)

where the – ve sign indicates that the work is done on the system by an external agent.

- The work done per unit charge in moving a test charge from point A to point B is the electrostatic potential difference between the two points(V_AB).

V_AB = W_A→B / Q

= - ∫_A^B(E . dl)

= - ∫_Initial^Final (E . dl)

- If the potential difference is positive, there is a gain in potential energy in the movement, external agent performs the work against the field.

- If the sign of the potential difference is negative, work is done by the field.

- The electrostatic field is conservative i.e. the value of the line integral depends only on end points and is independent of the path taken.

- Since the electrostatic field is conservative, the electric potential can also be written as:

V_AB = - ∫_A^B (E . dl )

= - ∫_A^P_o (E . dl) - ∫ _{P_o}^B (E . dl)

= - ∫_{P_o}^B (E . dl) - (- ∫_A^P_o(E . dl)

= V_B – V_A

Thus the potential difference between two points in an electrostatic field is a scalar field that is defined at every point in space and is independent of the path taken.

- The work done in moving a point charge from point A to point B can be written as:

W_A→B = - Q [V_B – V_A] = - Q ∫_A^B (E . dl)

- Consider a point charge Q at origin O.

- Now if a unit test charge is moved from point A to Point B, then the potential difference between them is given as:

- Electrostatic potential or Scalar Electric potential (V) at any point P is given by:

V = - ∫_{P_o}^P (E . dl)

The reference point P_o is where the potential is zero (analogues to ground in a circuit).

- The reference is often taken to be at infinity so that the potential of a point in space is defined as

V = - ∫_∞^P (E . dl)

- Basically potential is considered to be zero at infinity. Thus potential at any point ( r_B = r) due to a point charge Q can be written as the amount of work done in bringing a unit positive charge from infinity to that point (i.e. r_A → ∞)

- Electric potential (V) at point r due to a point charge Q located at a point with position vector r₁ is given as:

- Similarly for N point charges Q₁, Q₂ ….Q_n located at points with position vectors r₁, r₂, r₃…..r_n, the electric potential (V) at point r is given as:

- The charge element dQ and the total charge due to different charge distribution is given as:

dQ = ρ_ldl    → Q = ∫_L (ρ_ldl) → (Line Charge)

dQ = ρ_sds   → Q = ∫_S (ρ_sds) → (Surface Charge)

dQ = ρ_vdv    → Q = ∫_V (ρ_vdv) → (Volume Charge)

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Field Theory - Numericals/Solved Examples - Gauss's law...

2010-01-13T18:42:00.008+05:30

Q.1 A point charge of 30 nC is located at the origin while plane y = 3 carries charge 10 nC / m² . Find D at (0, 4, 3)?
Ans:

Electric flux density (D) at point (0, 4, 3) due to a point charge and line charge is given as:

D = D_Q + D_ρ

= 0.019 (0, 4, 3) + 5a_y

= 0.76 a_y + 0.057 a_z + 5a_y

= 5.076 a_y + 0.057 a_z

Q.2 A charge distribution in free space has ρ_v = 2r nC / m³ for 0 < r < 10 m and zero otherwise. Determine E at r = 2m and r = 12m?
Ans:

Gauss’s law states that:

For 0 ≤ r ≤ 10

D_r (4πr²) = ∫ ∫ ∫ 2r (r² sinθ dr dθ dφ)

D_r (4πr²) = 4π (2r⁴) / 4

For r ≥ 10

D_r (4πr²) = ∫ ∫ ∫ 2r_o (r² sinθ dr dθ dφ)

D_r (4πr²) = 4π (2r_o⁴) / 4

D_r (4πr²) = 2πr_o⁴

Q.3 If D = (2y² + z)a_x + 4xy a_y + x a_z C/m², find
a) Volume charge density at (-1, 0, 3)
b) The flux through the cube defined by 0 ≤ x ≤ 1,
c) The total charge enclosed by the cube.
Ans:

a) Volume charge density (ρ_v) is defined as:

ρ_v = ∇ . D

= 4

At point (-1, 0,3) ρ_v = 4

b) The total charge (Q) enclosed by the cube

Q = ∫_v ρ_v dv

= ∫_(x=0)¹ ∫_(y=0)¹ ∫_(z=0)¹ 4x (dxdydz)

= 4 | (x² / 2) |₀¹ (1) (1)

= 2 C

c) The flux through the cube defined by:

Ψ = Q_enc = 2C

Q.4 If the volume charge density (ρ_v) of a given charge distribution is given by ρ = ρ_o (a / r) in spherical co-ordinate, determine the electric flux density (D) at any point ?

Ans:

Gauss’s law states that:

Q_enc = ∫_v ρ_v dv

= ∫(r=0)^r ∫_(θ=0)^π ∫_(φ=0)^2π ρ_o (a/r) (r² sinθ dr dθ dφ)

= - ρ_o a | r² / 2 |₀^r (2π) | (cosθ) |₀^π

= - ρ_o a (r² / 2) (2π) ( -2)

= 2 ρ_o aπr²

Electric flux is given as:

Since,

Ψ = Q_enc

D_r (4πr²) = 2 ρ_o aπr²

D = [ (ρ_o a) / 2a ] a_r

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- Scalar Electric Potential / Electrostatic Potential (V)...

- Relationship Between Electric Field Intensity(E) and Electric Potential(V)...

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Field Theory - Gauss Law - Application To A Uniformly Charged Sphere...

2010-01-12T18:10:00.007+05:30

- Consider a sphere of radius “a” having a uniform volume charge density of ρ_o C/m³.

- Gaussian surface selected for a symmetric sphere charge is a sphere itself.

- Here we consider two cases:
a) A Gaussian surface with a radius r < a.

b) A Gaussian surface with a radius r > a.

CASE 1:

If r < a (consider the 1st figure), the total charge enclosed by the Gaussian surface is:

Electric flux is given as:

Here ds is taken to be a function of θ and φ( since the surface is a hollow sphere).

Gauss law states that Ψ = Q_enc

Therefore,

D_r 4π r² = (4 / 3) (ρ_o πr³)

Or

D = (r / 3) ρ_o a_r (0 < r < a)

CASE II:

If r > a (consider the above figure), the total charge enclosed by the Gaussian surface is:

Electric flux is given as:

Gauss law states that Ψ = Q_enc

Therefore,
D_r 4π r² = (4 / 3) (ρ_o πa³)

Or

D = (a³ / 3r² ) ρ_o a_r (r > a)

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Field Theory - Gauss Law - Application To An Infinite Line & Sheet Charge...

2010-01-12T12:52:00.011+05:30

- Consider an infinite line charge carrying a charge per unit length of ρ_L along the z axis.

- Gaussian surface selected for a symmetric line charge is a hollow cylinder of radius ρ and length l as shown in the figure

- A cylinder has basically three surfaces: top, bottom and the curved cylindrical surface.

From the diagram its clear that

- Electric flux density (D) is parallel to the top and bottom Gaussian surface.

- Electric flux density (D) is normal to the curved cylindrical Gaussian surface.

- Differential surface (ds) is always normal to a surface.

D and ds are normal to each other for the top and bottom Gaussian surface.

Hence

∫ (D . ds) = 0

Hence it’s clear that D and ds are parallel to each other only for the curvilinear Gaussian surface.

- Since the Gaussian surface is a hollow cylinder, hence the variable terms are φ and z. Thus the differential surface for a hollow cylinder is given as:

ds = ∫_(φ=0)^2π ∫₀^l (ρ dφ dz) a_ρ → S = 2πρl a_ρ

Hence applying gauss law, we have

→ Q_enc = D_ρ 2πρl = ρ_L l

INFINITE SHEET OF CHARGE

- Consider an infinite sheet charge carrying a charge per unit surface of ρ_s on a x-y or z = 0 plane.

- Gaussian surface selected for a symmetric sheet charge can be either a cylinder box or a rectangular box.

- The Gaussian surface is placed such that two of its faces are parallel to the sheet.

- In this case say the Gaussian surface is a rectangular box.

- From the diagram it’s very clear that only two faces of the rectangular box is parallel to the sheet and also parallel to the z-axis.

- Applying Gauss law, we have

- The electric field(E) as well as electric flux density(D) both points away from the plane if ρs is positive and towards the plane if ρs is negative.

- The magnitude of the electric field is a constant – the magnitude is independent of the distance from the infinite plane.

- This is because no matter how far the point is from the infinite sheet, the distance becomes incomparable with the dimensions of the plane. Hence it seems the point is very close to the infinite plane.

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Field Theory - Gauss Law (Theory) & Application To A Point Charge...

2010-01-12T00:01:00.007+05:30

- Gauss law is one of the fundamental laws of Electrostatics.

- It states that

“The net electric flux emanating or coming from a close surface S is equal to the total charge contained within the volume V bounded by that surface.”

We know,

Therefore

Ψ = Q_enc

Hence Gauss law also states that

“The total electric flux Ψ through any closed surface is equal to the enclosed electric charge.”

- Charge contained in a volume is given as:
dQ = ρ_v dv → Q = ∫_V ρ_v dv

Hence we have,

Applying Divergence theorem we have,

Comparing the above two equations, we have

And hence

∇ . D = ρ_v

This equation is called the 1^st Maxwell's equation of electrostatics.

- Gauss law is an easy way of finding electric field for some symmetric problems in electrostatics.

- Gauss law relates the electric field at points on a closed Gaussian surface to the net charge enclosed by that surface.

- One thing to remember is Q_enc contains charges which are enclosed within the volume. Charges outside the volume, no matter how large or how close it may be, are not included in the term Q_enc.

Procedure To Apply Gauss Law:

- Check out whether symmetric charge distribution exists or not.

- If yes, a hypothetical surface called a Gaussian surface is selected such that E / D (due to charge) is either normal or tangential to the Gaussian surface.

- Gaussian surface is a hypothetical (any imaginary) closed surface enclosing the charge configuration.

- Gaussian surface is a surface to which the electric flux density is normal and over which equal to a constant value.

Application Of Gauss Law To A Point Charge:

- Consider a point charge Q at the origin, say at point P, the electric flux density (D) is to be evaluated.

- Gaussian Surface selected for a point charge is a sphere centered at origin.

From the diagram its clear that

- Electric flux density is everywhere normal to the Gaussian surface i.e. D = D_r a_r

- The total charge enclosed by the Gaussian surface, Q_enc = Q.

- Since the Gaussian surface is a hollow sphere, hence the variable terms are θ and φ. Thus the differential surface for a hollow sphere is given as:

ds = ∫_(φ=0)^2π ∫_(θ=0)^π (r² sinθ dθ dφ) a_r = 4πr² a_r

- Hence applying gauss law, we have

→ Q_enc = D_r 4πr²

Hence

Again we know that, D = ε E.

Therefore

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- Gauss Law - Application To A Uniformly Charged Sphere...

- Numericals / Solved Examples - Gauss's law...

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Field theory - Numericals - Electric field Intensity (E) - Line, Surface and Mixed Charge Configuration...

2010-01-11T19:00:00.009+05:30

Q.1 An infinite long uniform line charge is located at y = 3, z = 5. If ρ_L = 30 nC/m. Calculate the electric field at point (5, 6, 1).
Ans:

The electric field intensity (E) due to an infinite line charge is given as:

In this case, ρ = (6 – 3) a_y + ρ(1 – 5)a_z = 3a_y – 4a_z

Hence a_ρ = (3a_y - 4a_z) / 5

Therefore E is,

Q.2 A uniform line charge density 20 nC lies on z axis between z = 1 and z = 3.
Find electric field intensity at point (4, 0, 0).
Ans:

Electric field due to a line charge density is given as:

In this case

R = 4 a_x – z a_z

a_r = (4 a_x – z a_z) / (4² + z²)

Hence we have,

Q. 3 Two identical charges are located on the y axis at y = 3 and y = 9. At what point in space is the net electric field zero?
Ans:
As the both charges are on the y axis, the point at which the fields due to the two charges can cancel has to lie on the y-axis also. Since the two charges are identical, the point at which the net electric field will be zero is midway between them, i.e. at point (0, 6, 0)

Q.4 A point charge 100 pC is located at (4, 1, -3) while the x-axis carries charge 2 nC/m. If the plane z = 3 also carries charge 5nC / m², find E at (1, 1, 1).
Ans:
Electric field at point (1, 1, 1) is given as:

= (- 0.0216, 0, 0.0288) + (0, 18, 18) – 264.7 (0, 0, 1)

= - 0.0216 a_x + 18a_y – 264.7 a_z V/m

Q.5 Plane x + 2y = 5 carries charge ρ_s = 6 nC/m². Determine E at (-1, 0, 1)?
Ans:

Let f (x, y) = x + 2y – 5

∇ f = a_x + 2a_y
A unit normal vector to any surface or plane is given as:

Now since (-1, 0, 1) lies below the plane, therefore Electric field is given as:

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- Gauss's Law - Theory.

- Gauss's Law - Application To a Point charge.

- Gauss's Law - Application To An Infinite Line Charge.

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Field Theory - Numericals - Electric field Intensity (E) and Force (F)

2010-01-11T18:23:00.005+05:30

Q.1 Point charges Q₁ = 5 μC and Q₂ = -4 μC are placed at (3, 2, 1) and (-4, 0, 6) respectively. Determine the force on Q₁.
Ans:
Force on Q₁ due to Q₂ is given as :

= -5.74 a_x – 1.642 a_y + 4.104 a_z mN

Q. 2 Five Identical 15 μC point charges are located at the center and corners of a square defined by -1 < x, y > 1 and z = 0.
Find the force on the 10 μC point charge at (0, 0, 2).

Therefore,
F / 1.35 = [(0, 0, 2) / 8] + [(-1, -1, 2) / 6^3/2] + [(1, -1, 2) / 6^3/2]
             + [(-1, 1, 2) / 6^3/2] + [(1, 1, 2) / 6^3/2]

→ F = 1.35 [(0.25 + (8 / 6^3/2)] a_z

        = 1.072a_z N

Q.3 Point Charges Q₁ and Q₂ are respectively located at (4, 0, -3) and (2, 0, 1). If Q₂ = 4nC, find Q₁ such that
a) The E at (5, 0, 6) has no z component.
b) The force on a test charge at (5, 0, 6) has no x component.

a) Electric field intensity at point (5, 0, 6) is given as:

Hence Q1 = - 3.463 nC.

b) F (5, 0, 6) = qE (5, 0, 6)

If E_x = 0

Hence, Q1 = -18.7 nC.

Q. 4 Determine the total charge
a) On line 0 < x < 5 m if ρ_L = 2x² mC/m
Ans:

Q = ∫_L ρ_l dl

= ∫₀⁵(2 x²)dx

= | 4x³ |₀⁵ = 0.5 C

b) On the cylinder ρ = 3, 0 < z < 4, if ρ_s = ρz² nC/m²∫_Sρ_sds = ∫_S ρz² ds

= ∫_Sρz² (ρ dφ dz)

= ∫_{(φ =0)}^2π ∫_{(z =0)}⁴ ρz² (ρ dφ dz)

= (ρ³ / 3) (2π) | z³ / 3 |₀⁴

= 9 x (2π) x (4³ / 3)

= 1.206 μC

c) Within the sphere r = 4m, if ρ_v = 10 / (r sinθ ) C / m³
Ans:
Q = ∫_V ρ_vdv

= ∫_V [ 10 / (r sinθ ) ] (r² sinθ dr dθ dφ)

= ∫_(r=0)⁴ ∫_(θ=0)^π ∫_(φ=0)^2π (10 rdr dθ dφ)

= 10 | r² / 2 |₀⁴ (2π) (π)

= 157.91 C

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- Numericals - Electric field Intensity (E) - Line, Surface and Mixed Charge Configuration...

- Gauss Law (Theory) & Application To A Point Charge...

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Field Theory - Electric Field Intensity (E) Due To a Circular Disk Charge...

2010-01-09T16:21:00.006+05:30

- Consider a circular disc of radius ‘a’ which carries a uniform surface charge density ρ_s , C /m².

- Say the disk lies on x-y plane (or z = 0 plane) with its axis along the z axis as shown in the figure.

- We need to find out electric field (E) due to a circular disk at a point P (0, 0, h) on the z axis (z > 0).

- Electric field intensity (E) at a point due to any surface charge (ρ_s) is given as:

-Consider the triangle shown in figure(Since it’s a disc, the varying terms are radius ρ and angle φ)

As per the vector law of addition,

ρ a_ρ + R = h a_z → R = - ρ a_ρ + h a_z

| R | = (ρ² + h²)^1/2

a_R = R / | R |

a_R = - ρ a_ρ + h a_z / (ρ² + h²)^1/2

Substituting all these values in the above equations, the electric field intensity E becomes

- Contribution along a_ρ due to symmetry adds up to zero.

- Therefore the final electric field intensity at point (0, 0, h) has only z component.

- As a → 0, the electric field intensity (E) also tends to zero i.e. E → 0

- Hence electric field intensity (E) at point (0, 0, h) is given as:

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- Numericals - Electric field Intensity (E) and Force (F)...

- Numericals - Electric field Intensity (E) - Line, Surface and Mixed Charge Configuration...

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Field Theory - Electric Field Intensity (E) Due To a Circular Ring Charge...

2010-01-08T12:52:00.007+05:30

- Consider a circular ring of radius 'a' which carries a uniform line charge density ρ_L as shown in figure.

- We need to find out electric field at a point P (0, 0, h) on the z axis (z > 0).

- Electric field intensity (E) due to any line charge (ρ_L) in general is given as:

In this case,

dl = a dφ (Since the differential part dl is a differential arc)

R² = a² + h²

Consider the triangle shown in the above figure

a a_ρ + R = h a_z → R = - a a_ρ + h a_z

a_R = R / | R |

a_R = - a a_ρ + h a_z / ( a² + h² )^1/2

Substituting all these values in the above equations, the electric field intensity E becomes:

- For every element dl there is a corresponding element diametrically opposite that gives an equal but opposite dE_ρ so that the two contributions cancel each other.

Hence contribution along a_ρ due to symmetry adds up to zero.

- Therefore the final electric field intensity at point (0, 0, h) has only z component.

- Hence Electric field intensity (E) due to a circular ring (of radius a carrying a uniform charge ρ_L) placed on the x-y plane and if the point of interest is any point on z axis, then it is given as:

- Similarly if the ring is placed on x-z plane and point of interest is any point on y- axis, then E is given as:

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- Numericals - Electric field Intensity (E) and Force (F)

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Field Theory - Electric Field Intensity Due To a Infinite Sheet Charge....

2010-01-05T11:11:00.005+05:30

(SYMMETRY ABOUT A PLANE)

- Consider an infinite sheet of charge in the y-z plane having a uniform charge density of ρ_s C/m².

- For easy analysis, divide the sheet into differential width strips (dy).

ρ_L = ρ_s dy

- The distance between the point P and the line charge

Z² = X² + Y²

- Since the sheet lies in y-z plane the field components due to y and z will be canceled out at the point at which the field is to be determined.
(As we have seen in electric field due to a line charge where the z component gets canceled out.)

- Only E_x component is present and hence the field is a function of x alone for an infinite sheet of charge on y-z plane.

- Hence the electric field intensity due to an infinite line charge is given as:

- Magnitude E_x at point P due to an infinite differential width strip(dy):

from – ∞ to + ∞ ,

- If the field intensity is obtained at point P on the negative axis, then E will be:

- In general, Electric field intensity for an infinite sheet of charge is given as:

Where a_n is a unit vector normal to the sheet.

- The electric field intensity (E) points away from the plane if ρ_s is positive and towards the plane if ρ_s is negative.

- The magnitude of the electric field is a constant – the magnitude is independent of the distance from the infinite plane.

- This is because no matter how far the point is from the infinite sheet, the distance becomes incomparable with the dimensions of the plane. Hence it seems the point is very close to the infinite plane.

- In a parallel plate capacitor the electric field intensity between the two plates having equal and opposite charge is given by:

The first –ve sign denotes –ve charge on one plate and the second –ve sign denotes opposite direction.

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- Electric Field Intensity (E) Due To a Circular Ring Charge...

- Electric Field Intensity (E) Due To a Circular Disk Charge...

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Field Theory - Electric Field Intensity Due To a Finite Line Charge...

2010-01-04T00:06:00.005+05:30

- Consider a line charge with uniform charge density ρ_L extending from + a to – a along the z- axis.

- Charge element dQ associated with the element dl of the line is

dQ = ρ_L dl = ρ_L dz

- From the diagram it’s clear that the Electric field intensity has two components i.e. E_ρ and E_z.

- If we move around the line charge and if we vary ρ, while keeping φ and z constant, it is expected that the field would become weaker as ρ increases.

- No element of charge produces a φ component i.e. E_φ is equal to zero.

- Hence Electric field intensity at point P has only two component one along the ρ and the other along the z direction.

- However the contribution due to E_z component by the elements of charge will be canceled because the same are at equal distances above and below the point at which the field is to be determined.

(For Example: If a charge element is present at + 4a_z and another charge element at - 4a_z, then the z component due to the two mentioned charge element at the point P will be canceled out on resolving the field.)

- Hence the only field component that exist at point P due to a symmetric line charge element is E_ρ.

- Differential electric field intensity(dE) due to a small line charge element is given as:

- In this case r is equivalent to l (small L). Therefore the equation can be written as:

- Consider the triangle OPQ, we have

(Small L) l = (z² + ρ²)^1/2

cos θ = ρ / l ; sin θ = z / l

dE_ρ = |dE| cosθ = dE . (ρ / l)

Similarly we have,

dE_z = |dE| sinθ = dE . (z / l)

Since the resultant z component E_z of the field at a point on the ρ axis is zero i.e. ∫dE_z = 0

The axial component E_ρ of the field at a point P on ρ axis can be obtained by integrating from – a to + a as

∫ dE_ρ = ∫_-a^+a dE . (ρ / l)

- For an infinite length of line charge “a → ∞"

- Hence the electric field intensity due to an infinite line charge is given as:

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- Electric Field Intensity Due To a Infinite Sheet Charge.

- Electric Field Intensity Due To a Circular Ring Charge.

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Field Theory - Electric Field Strength / Electric field Intensity (E)...

2010-01-03T12:12:00.008+05:30

- Electric field due to a charge is the space around the unit charge in which it experiences a force.

- Electric field intensity or the electric field strength at a point is defined as the force per unit charge.

Mathematically,

E = F / Q

F = E Q

- The force on charge Q is the product of a charge (which is a scalar) and the value of the electric field (which is a vector) at the point where the charge is located.

- Hence force will be either parallel or anti-parallel to the Electric field intensity.
(i.e. Q > 0) the force F points in the same direction as the electric field E.

- If the charge is negative (i.e. Q < 0) the force F points in the opposite direction as the electric field E.

- Electric field intensity(E) at point r due to a point charge Q located at a point with position vector r₁ is given as:

Similarly for N point charges Q₁, Q₂ ….Q_n located at points with position vectors r₁, r₂,….r_n, the electric field intensity at point r is given as:

- Charges can occur as point charge, line charge, surface charge and volume charge.

The charge element dQ and the total charge due to different charge distribution is given as:

dQ =ρ_ldl →   Q = ∫_L ρ_ldl       (Line Charge)

dQ = ρ_sds →   Q = ∫_S ρ_sds      (Surface Charge)

dQ = ρ_vdv →   Q = ∫_V ρ_vdv     (Volume Charge)

- Electric field intensity due to different charge distribution is hence given as:

ELECTRIC LINES OF FORCES:

- Electric line of force is a pictorial representation of the electric field.

- Electric line of force (also called Electric Flux lines or Streamlines) is an imaginary straight or curved path along which a unit positive charge tends to move in an electric field.

PROPERTIES OF ELECTRIC LINES OF FORCE:

- Lines of force start from positive charge and terminate either at negative charge or move to infinity.

- Similarly lines of force due to a negative charge are assumed to start at infinity and terminate at the negative charge.

- Lines of force never intersect i.e. they do not cross each other.

Tangent to a line of force at any point gives the direction of the electric field E at that point.

- Lines are dense close to a source of the electric field and become sparse when one moves away.

-The number of lines per unit area, through a plane at right angles to the lines, is proportional to the magnitude of E. This means that, where the lines of force are close together, E is large and where they are far apart E is small.

IMPORTANT POINTS:

- If there is no charge in a volume, then each field line which enters it must also leave it.

- If there is a positive charge in a volume then more field lines leave it than enter it.

- If there is a negative charge in a volume then more field lines enter it than leave it.

- Hence we say

“Positive charges are sources and Negative charges are sinks of the field.”

ELECTRIC FLUX DENSITY(D = εE):

- Line of Force may be termed as ‘Electric Flux’ represented by ψ and unit is coulomb (C).

- The density of electric flux is the electric (displacement) flux density, D.

- It is the measure of cluster of ‘electric lines of force’. It is the number of lines of force per unit area of cross section.

D = ψ / S → ψ = ∫_S(D . ds)

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- Electric Field Intensity Due To a Finite Line Charge...

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Field Theory - Introduction to Electrostatics and Coulomb's Law...

2010-01-02T23:08:00.013+05:30

- Electrostatics is a branch of science that involves the study of various phenomena caused by electric charges that are slow-moving or even stationary.

- Electric charge is a fundamental property of matter and charge exist in integral multiple of electronic charge.

Electrostatics as the study of electric charges at rest.

- The two important laws of electrostatics are
          - Coulomb’s Law.
          - Gauss’s Law.

- Both these laws are used to find the electric field due to charge configurations.

- Coulomb’s law is applicable in finding electric field due to any charge configurations where as Gauss’s law is applicable only when the charge distribution is symmetrical.

COULOMB’S LAW:

- Coulomb’s law is the “Law of Action” which describes the force between two point charges.

- A point charge is a charge that occupies a region of space which is negligibly small compared to the distance between the point charge and any other object.

- Coulombs law states that “ The force of attraction or repulsion between two point charges Q1 and Q2 is:

          - Proportional to the charges Q1 and Q2.
          - Varies inversely as the square of distance between them.
          - Acts along the line joining the two point charges.

- Mathematically, Coulomb’s law is expressed as:

Where k is the proportionality constant and is defined as

ε_o is known as the permittivity of free space and ε_r is called the relative permittivity of any dielectric material.

In Free Space, ε_r = 1, hence

Hence equation of force in free space becomes:

- Here the diagram represents the coulomb
vector force on Point charges Q1 and Q2.

Force F₁₂ on Q₂ due to Q₁ is given as:

- Similarly F₂₁ on Q₁ due to Q₂ is given as:

F₂₁ = |F₁₂ | a_R21 = |F₁₂ | ( - a_R12 )

F₂₁ = - F₁₂ since [a_R21 = - a_R12 ]

- The force on Q₁ due to Q₂ is equal in magnitude but opposite in direction to the force on Q₂ due to Q₁.

- If there are more than two point charges, then the principle of Superposition can be used to determine the force on a particular charge.

If there are N numbers of charges Q₁, Q₂, Q₃…..Q_n located respectively at points with position vectors r₁, r₂, r₃….r_n, the force experienced by a charge Q located at position vector r is given as:

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- Electric Field Strength / Electric field Intensity (E)...

- Electric Field Intensity Due To a Finite Line Charge...

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Field Theory - ELECTROMAGNETIC THEORY MADE EASY........

2009-12-22T20:48:00.016+05:30

INDEX

VECTOR ALGEBRA:

- Scalars and Vectors.

- Unit Vectors.

- Position and Distance Vectors.

- Vector Multiplication.

- Components Of a Vector.

- Numericals / Solved Examples.

COORDINATE SYSTEMS & TRANSFORMATION:

- Introduction To Coordinate System.

- Cartesian Coordinate System / Rectangular Coordinate System (x, y, z).

- Differential Analysis Of Cartesian Coordinate System.

- Circular Cylindrical Coordinate System (ρ, φ, z).

- Differential Analysis Of Cylindrical Coordinate System.

- Spherical Coordinate System ( r, θ , φ).

- Differential Analysis Of Spherical Coordinate System.

- Numericals / Solved Examples - Page 1.

- Numericals / Solved Examples - Page 2.

VECTOR CALCULUS:

- Line , Surface and Volume Intergral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Numericals / Solved Examples - Gradient Of a Scalar.

- Divergence Of a Vector ( ∇ . A ).

- Numericals / Solved Examples - Divergence Of a Vector.

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇² V).

ELECTROSTATIC FIELD:

- Introduction To Electrostatics.

- Coulomb's law.

- Electric Field Intensity (E).

- Electric Lines Of Forces /Streamlines / Electric Flux (ψ) .

- Electric Flux Density (D).

- Electric Field Intensity Due To a Finite and Infinite Line Charge.

- Electric Field Intensity Due To a Infinite Sheet Charge.

- Electric Field Intensity Due To a Circular Ring Charge.

- Electric Field Intensity Due To a Circular Disk Charge.

- Numericals / Solved Examples - Electric Force and Field Intensity.

- Numericals / Solved Examples - Electric Field Intensity - Line, Surface and Mixed Charge Configuration.

- Gauss's Law - Theory.

- Gauss's Law - Application To a Point charge.

- Gauss's Law - Application To An Infinite Line Charge.

- Gauss's Law - Application To An Infinite Sheet Charge.

- Gauss's Law - Application To a Uniformly Charged Sphere.

- Numericals / Solved Examples - Gauss's Law.

- Scalar Electric Potential / Electrostatic Potential (V).

- Relationship Between Electric Field Intensity (E) and Electrostatic Potential (V).

- Electric Potential Due To a Circular Disk.

- Electric Dipole.

- Numericals / Solved Examples - Electric Potential and Electric Dipole.

- Energy Density In Electrostatic Field / Work Done To Assemble Charges.

- Numericals / Solved Examples - Electrostatic Energy and Energy Density.

ELECTRIC FIELD IN MATERIAL SPACE:

- Properties Of Materials.

- Current (I) and Current Density (J).

- Conduction and Convection Current Density.

- Isolated Conductor Under The Influence Of An Applied Electric Field (E).

- Conductor Wired To a Source Of Electromotive Force.

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Field Theory - Laplacian of a Scalar - Defination, Significance and Solved Examples...

2009-12-19T23:53:00.009+05:30

- The laplacian is a scalar operator. Hence when applied to scalar field, the result is also a scalar field.

- Laplace operator is a second order differential equation defined as the divergence of the gradient of scalar field.

- Laplacian of a scalar field V is written as ∇² V.

- Laplacian of a scalar is:
            - Used in electrostatics to represent the charge associated to a given potential
            - Used in defining the Helmholtz equation of propagation of EM wave.
            - Used in Laplace’s and Poisson’s equation.

Laplacian of a scalar field V in Cartesian coordinate system is given as:

Laplacian of a scalar field V in Cylindrical coordinate system is given as:

Laplacian of a scalar field V in Spherical coordinate system is given as:

Q.1 Determine the Laplacian of the scalar fields:
a) V = x²y + xyz
Ans:

=> ∇² V = [(d / dx) ( 2xy + yz)] +[ (d / dy) ( x² + xz)] + [(d / dz) (xy)] = 2y

b) V = ρz sinφ + z²cos² φ + ρ²
Ans:

= (1 / ρ) (z sinφ + 4ρ) – (1 / ρ²) (2ρ sinφ + 2z² cos 2φ) + 2 cos² φ

= 4 + 2 cos² φ - (1 / ρ²) 2z² cos 2φ

- ... Back To Index.

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Field Theory - Curl Of a Vector Field ( curl A) - Defination, Significance and Solved Examples...

2009-12-19T22:44:00.010+05:30

Circulation of a vector field A around a closed path L is defined as:

Mathematically Curl of a vector A is defined as:

Where the area Δ S is bounded by the curve L and the unit vector a_n is the unit vector normal to the surface.

- The direction of the curl is the axis of rotation, as determined by the right hand thumb rule and the magnitude of the curl is the magnitude of rotation.

- From the above relation we can define Curl as the maximum circulation per unit area.

- The curl of a vector field provides another vector field that indicates rotational sources of the original vector field.

- Curl is a measurement of the circulation of vector field A around a particular point.

- If there’s a paper boat in a whirlpool, the circulation would be the amount of force that pushed it along as it went in a circle. The more circulation, the more pushing force you have.

- Consider a closed loop counter C. The circulation will be positive if a component of vector field A is pointing in the direction dl at every point on counter C. Hence if the circulation is positive then obviously the curl of a vector A will also be positive.

- Similarly if a component of vector field A points in the opposite direction (- dl) at every point of the counter, then the circulation and thus the curl will be negative.

- If the curl of a vector field A is zero, then the vector field A is said to be irrotational or potential (if ∇ x A =0). In such cases, the circulation of A around a closed path is zero; it implies that the line integral of A is independent of the chosen path. Hence an irrotational field is also known as a conservative field.

Curl of a vector A in Cartesian Coordinate system is given as:

The above is the determinant form of the formula for curl. The first line is made up of unit vectors, the second of scalar operators, and the third of scalar functions, so this is not a determinant in the strict mathematical sense.

Curl of vector A in Cylindrical coordinate system is given as:

Curl of vector A in Spherical coordinate system is given as:

STOKES THEOREM:

It states that the circulation of a vector field A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L.

The divergence theorem relates a closed surface integral to an open volume integral, the Stokes theorem relates a closed line integral to an open surface integral.

Q.1 Determine the curl of each of the vector field:
a) A = yz a_x + 4xy a_y + y a_z
Ans:

=> ∇ x A = a_x (1 – 0) + a_y (y – 0) +a_z (4y – z)

= a_x + y a_y + (4y – z) a_z

b) A = ρz sinφ a_ρ + 3ρz²cosφ a_φ
Ans:

=> ∇ x A = a_ρ ( 0 - 6ρzcosφ) + a_φ (ρsinφ – 0)
+ a_z (6ρz² cosφ – ρzcosφ)

= - 6 ρzcosφ a_ρ + ρsinφ a_φ + (1 / ρ)( 6ρz² cosφ – ρzcosφ) a_z

= - 6 ρzcosφ a_ρ + ρsinφ a_φ + (6z – 1) zcosφ a_z

Q.2 Show that A = (y + z cos xz) a_x + x a_y + xcosxz a_z is conservative, without computing any integrals?
Ans:
If A is conservative, then curl of vector A should be equal to zero.

=>∇ x A = 0.
Hence A is a conservative field.

- ... Back To Index.

NEXT TOPIC:

- Laplacian Of a Scalar ( ∇² V).

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Field Theory - Numericals / Solved Examples - Divergence of a Vector...

2009-12-19T18:07:00.009+05:30

Q.1 Determine the divergence of the following vector fields and evaluate them at the specified points:
a) A = yz a_x + 4xy a_y + y a_z at point (1, -2, 3)
Ans:

Given A = yz a_x + 4xy a_y + y a_z at point (1, -2, 3)

Divergence of A is given as:

∇ . A = 0 + 4x + 0 = 4x
At Point (1, -2, 3) Hence, ∇ . A = 4

b) A = ρzsinφ a_ρ + 3ρz² cosφ a_φ at (5, π / 2, 1)
Ans:

Divergence of a vector field A in cylindrical co-ordinate system is given as:

=> ∇ . A = (1/ρ) 2ρz sinφ - (1/ρ) 3ρz² sinφ

= 2zsinφ - 3z² sinφ = (2 - 3z) zsinφ

At point (5, π / 2, 1) ∇ . A = (2 – 3) (1) = -1

c) A = 2r cosθ cosφ a_r + r^1/2 a_φ at (1, π / 6, π / 3)
Ans:

Divergence of a vector field A in Spherical co-ordinate system is given as:

=> ∇ . A = (1/r²) 6r² cosθ cosφ = 6 cosθ cosφ

At point (1, π/6, π/3)

∇ . A= 6 cos π/6 cos π/3 = 2.598.

Q. 2 Determine the flux of D = ρ² cos²φ a_ρ + zsin φ a_φ over the closed surface of the cylinder, 0 < z < 1, ρ =4. Verify the divergence theorem for this case.
Ans:

We know a closed cylinder has three surface - top(t), bottom(b) and curved surface(s).

Ψ = Ψ_t + Ψ_b + Ψ_s

Ψ_t = Ψ_b = 0, since D has no z component.

It means,
Ψ_s = ∫ ∫ ρ² cos²φ (ρ dφ dz)
= ρ³ ∫ ₀^2πcos²φ dφ ∫ ₀¹dz
= (4)³ (π) (1) = 64 π

Therefore,

Ψ = Ψ_t + Ψ_b + Ψ_s = 64 π

By the divergence theorem,

Divergence of D is given as:

∇ . A = 3ρ cos²φ - (1/ρ) cosφ

= ∫ ∫ ∫(3ρ cos²φ - (1/ρ) cosφ) ρdρ dφ dz

= 3 ∫₀⁴ ρ²dρ ∫₀^2π cos²φ dφ ∫₀¹dz + ∫₀⁴dρ ∫₀^2π cosφdφ ∫₀¹ dz

= 3 (4³ /3 ) (π ) (1) = 64π

- ... Back To Index.

NEXT TOPIC:

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇² V).

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Field Theory - Divergence Of a Vector Field ( div A) - Defination and Significance...

2009-12-17T13:05:00.013+05:30

Mathematically divergence of a vector A is defined as:

Where the surface S is a closed surface that completely surrounds a small volume Δ v and where ds points outward from the closed surface.

- A divergence is applied to a vector as a function of position, yielding a scalar. The divergence actually measures how much the vector function is spreading out.

- Divergence of a vector field A is a measure of how much a vector field converges to or diverges from a given point. In simple terms it is a measure of the outgoingness of a vector field.

- Divergence of a vector field is positive if the vector diverges or spread out from a given point. If you are at a location from which the vector field tends to point away in all directions, you will definitely have a positive divergence. It means divergence is positive at a source point in the field.

- Divergence of a vector field is negative if the vector field converges at that point. If the field points inward toward a point, the divergence in and near that point is negative. It means divergence is negative at sink point in the field.

- Hence a nonzero divergence at some point means there must be a source or sink at that position.

- If just as much of the vector field points in as out, the divergence will be approximately zero.

- A vector field with constant zero divergence is called solenoidal or divergenceless or incompressible (∇ . A = 0). In such cases no net flow can occur across any closed surface.

- Divergence of a vector field results in a scalar field that represents the sources of the vector field.

- Divergence of a vector field A in Cartesian coordinate system is given as:

- Divergence of a vector field A in Cylindrical coordinate system is given as:

- Divergence of a vector field A in Spherical coordinate system is given as:

DIVERGENCE THEOREM:

- It states that the net outward flux of a vector field A through a closed surface S is equal to the volume integral of the divergence of the field A inside the surface.

- It also states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.

- ... Back To Index.

NEXT TOPIC:

- Numericals / Solved Examples - Divergence Of a Vector.

- Curl Of a Vector ( ∇ x A).

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Field Theory - Numericals / Solved Examples - Gradient of a scalar

2009-12-16T18:05:00.014+05:30

Q. 1 Find the gradient of these scalar fields:
a) U = 4xz₂ + 3yz
b) H = r²cosθ cosφ
Ans:
a)

=>∇ U = 4z² a_x + 3z a_y + (8xz +3y) a_z

b)

=>∇ H = 2r cosθ cosφ a_r + r sinθ cosφ a_θ + rcosθ sinφ a_φ

Q.2 If V_{(x, y, z)} = 3x²y –y²z², find ∇ V and |∇ V| at the point (1, 2, -1).
Ans:

∇ V = 6xy a_x + (3x² – 2yz²) a_y + (-2y²z) a_z

At the point (1, 2, -1)

∇ V = 6(1)(2) a_x + [3(12) – 2(2)(-1)²] a_y – 2(2)² (-1) a_z
=>∇ V = 12a_x - a_y + 8a_z

|∇ V|_{(1, 2, -1)} = |12a_x - a_y + 8a_z | = (209)^1/2

Q.3 Given φ = xy +yz +xz, find gradient φ at point (1, 2, 3) and thedirectional derivative of φ
Ans:
∇ φ = (y + z)a_x + (x + z)a_y + (y + z)a_z

At point (1, 2, 3)

∇ φ = 5a_x + 4a_y + 3a_z

The directional derivative is given as:

dφ/dl = ∇ φ . a_l
= (5, 4, 3) . [(3, 4, 4) – (1, 2, 3)] / 3
= [(5, 4, 3) . (2, 2, 1)] / 3 = 7

Q.4 Find V_{(x, y, z)} if grad V = (y² – 2xyz³)a_x + (3 + 2xy – x²z³)a_y + (4z³ – 3x²yz²)a_z and V_{(0, 0, 0)} = -2.
Ans:

∂V / ∂x = y² -2xyz³

∂V / ∂y = 3 + 2xy – x²z³

∂V / ∂y = 4z³ – 3x²yz²

V = ∫( y² – 2xyz³) dx = xy² – x²yz³ + f(y, z)

V = ∫( 3 + 2xy – x²z³) dy = 3y + xy² – x²yz³ + g(x, z)

V = ∫( 4z³ – 3x²yz²) dz = z⁴ – x²yz³ + h(x, y)

Comparing the above 3 equations, we have
f (x, y) = 3y + z⁴ + c

Hence,

V = xy² – x²yz³ + 3y + z⁴ + c

Since V_{(0, 0, 0)} = -2

V = xy² – x²yz³ + 3y + z⁴ – 2

Q.5 Find the unit normal vector of the surface x² + y² + z² = 14 at (-1, 3, 2) ?
Ans:
Here V _{(x, y, z)} = x² + y² + z²

∇ V = 2x a_x + 2y a_y + 2z a_z

At point (-1, 3, 2)

∇ V = -2a_x + 6a_y + 4a_z

|∇ V| = 2(14)^1/2

Unit normal vector
a_n = ∇ V / | ∇ V |
= - a_x + 3a_y + 2a_z / (14)^1/2

Q.6 The temperature in an auditorium is given by T = x² + y² – z. A mosquito located at (1, 1, 2) in the auditorium desires to fly in such a direction that it will get warm as soon as possible. In what direction must it fly?

Ans:

∇ T = 2xa_x + 2ya_y - a_z

At point (1, 1, 2)

∇ T = 2a_x + 2a_y - a_z

The mosquito should move in the direction of 2a_x + 2a_y - a_z.

- ... Back To Index.

NEXT TOPIC:

- Divergence Of a Vector ( ∇ . A ).

- Numericals / Solved Examples - Divergence Of a Vector.

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Field Theory - Gradient of a Scalar T (grad T) - Defination and Significance....

2009-12-15T15:08:00.016+05:30

- The gradient of a scalar field provides a vector field that states how the scalar value is changing throughout space – a change that has both a magnitude and direction.

- A gradient is applied to a scalar quantity that is a function of a 3D vector field: position. The gradient measures the direction in which the scalar quantity changes the most, as well as the rate of change with respect to position.

- The physical meaning of the gradient of a scalar is that it represents the steepness of the slope or line. For example, height is a scalar quantity; gradient of the height would be a vector pointing upwards. The length of the vector is proportional to the steepness of the slope.

- A derivative is required that tells us how fast the function varies, if we move a little distance.

- Consider a scalar function T which is a function of space coordinates x, y and z.

- The projection or the component of ∇ T in the direction of a unit vector a_l is ∇ T . a_l and is called the DIRECTIONAL DERIVATIVE of T along unit vector. Hence dT/dl is the directional derivative of T.

- Hence we also say that, the gradient of a scalar field indicates the direction of greatest change (that is largest derivative) as well as the magnitude of that change, at every point in space.

PROPERTIES OF GRADIENT OF A SCALAR FIELD T:

- If A =∇ T, T is said to be the scalar potential of A.

- The magnitude of ∇ T equals the maximum rate of change in T per unit distance.

- ∇T points in the direction of the maximum rate of change in V.

- ∇T at any point is perpendicular to the constant T surface that passes through that point.

Gradient of a scalar T for Cartesian coordinate system is given as:

Gradient of a scalar T for Cylindrical coordinate system is given as:

Gradient of a scalar T in Spherical coordinate system is given as:

- ... Back To Index.

NEXT TOPIC:

- Numericals / Solved Examples - Gradient Of a Scalar.

- Divergence Of a Vector ( ∇ . A ).

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Field Theory - Del Operator - Defination and Significance...

2009-12-15T14:23:00.010+05:30

- The collection of partial derivative operators is called DEL operator. Hence DEL can be viewed as the derivative in multi dimensional space.

- DEL operator is defined as a vector differential operator.

- A DEL operator is not a vector in itself, but when acts on a scalar function, it becomes a vector.

- Del is not simply a vector; it is a vector operator. Whereas a vector is an quantity with both a magnitude and direction, DEL does not have a precise value for either until it is allowed to operate on something.

- In Cartesian coordinate system Del operator is given as:

This operator is useful in defining
- Gradient of a scalar V (∇ V)
- Divergence of a vector A (∇ . A)
- Curl of a vector A (∇ x A)
- Laplacian of a scalar V (∇² V)

DEL OPERATOR - CYLINDRICAL CO-ORDINATE SYSTEM:

Unit vectors of Cartesian co-ordinate system are related to unit vectors of Cylindrical co-ordinate system as:

a_x = a_ρ cosφ – a_φ sinφ

a_y = a_ρ sinφ + a_φ cosφ

a_z =a_z

The differential part of x, y in terms of ρ and φ is given as:

Since the del operator is given as:

Substituting the values, we get

DEL OPERATOR - SPHERICAL CO-ORDINATE SYSTEM:

Unit vectors of Spherical co-ordinate system are related to unit vectors of Cartesian co-ordinate system as:

a_x = sinθ cosφ a_r + cosθ cosφ a_θ – sinφ a_φ

a_y = sinθ sinφ a_r + cosθ sinφ a_θ + cosφ a_φ

a_z = cosθ a_r - sinθ a_θ

The differential part of x, y, z in terms of r, θ and φ as:

Substituting the values, we get

- ... Back To Index.

NEXT TOPIC:

- Gradient Of a Scalar (∇ V).

- Numericals / Solved Examples - Gradient Of a Scalar.

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Field Theory - Line, Surface and Volume Integral...

2009-12-14T00:34:00.020+05:30

- The Line integral of Vector A along a path L is given as

∫_L A .dl

- The line integral is the dot product of a vector with a specified curve C.

- We can also say that line integral is the integral of the tangential component of vector A along the curve C.

- If the path of integration is a closed path, the line integral becomes a closed line integral and is called the circulation of A around C.

- Line Integral is useful in finding the electric field intensity along a path L.

- The surface integral of a vector B across a surface S is defined as

∫_s B .ds

- When the surface S is closed, the surface integral becomes the net outward flux of B across S, i.e.

- Surface integral is useful in finding the magnetic flux through a surface S.

- The volume integral of a scalar T over a volume v is given as

∫_v T . dv

Q.1 Calculate the circulation of A = ρ cosφ a_ρ+ z sinφ a_z around the edge L of the wedge defined by 0 < ρ < 2, 0 < φ < 60^o, z = 0 as shown.

A = ρcosφ a_ρ+z sinφ a_z

∫₁ A .dl = ∫₀² (ρcosφ a_ρ+zsinφ a_z) dρa_ρ = ρ² cosφ / 2 = 4 / 2 = 2 (since φ = 0^o)

∫₂ A .dl = ∫₀^π/3 (ρcosφ a_ρ + zsinφ a_z) ρdφa_φ = 0

∫₃ A .dl = ∫₂⁰(ρcosφ a_ρ+zsinφ a_z) dρa_ρ = - 4 cosφ / 2

= -1 (since φ = 60^o)

Q.2 Given that H = x² a_x + y² a_y, evaluate ∫_L H .dl where L is along the curve y =x² from (0, 0) to (1, 1).
Ans:
∫_L H .dl = ∫( x² a_x + y² a_y ) . (dx a_x + dy a_y + dz a_z)
= ∫( x² dx+ y² dy )

But on L, y = x² hence dy = 2x dx

Therefore
∫_L H .dl = ∫₀¹[x² dx+ x⁴ (2xdx) ] = ∫₀¹( x² dx+ 2x⁵ dx )

= | x³/3 |₀¹ + 2 | x⁶ /6 |₀¹ = 0.667

Q.3 Given that ρ_s = x² + xy, calculate ∫_s ρ_sds over the region y ≤ x², 0< x< 1.
Ans:

∫_sρ_sds = ∫ ∫ x² dxdy + ∫ ∫ xy dx dy

= ∫₀¹x² dx ∫dy + ∫₀¹x dx ∫ydy

= ∫₀¹x² dx | y | + ∫₀¹x dx | y² / 2 |

= ∫₀¹x⁴ dx + ∫₀¹ ( x⁵ / 2) dx = 1/5 + 1/12 = 0.2833

- ... Back To Index.

NEXT TOPIC:

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

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Field Theory - Numericals On Vector Algebra...

2009-12-07T11:14:00.015+05:30

Q.1. Given vectors A = a_x + 3a_z and B = 5a_x + 2a_y -6a_z, determine

a) |A+B|
b) 5A – B
c) Component of A along a_y.
d) Unit vector parallel to 3A + B.

Ans:

a) A + B = a_x +3a_z +5a_x +2a_y - 6a_z = 6a_x + 2a_y – 3a_z.

|A + B| = (6² + 2² + 3²)^1/2 = 7.

b) 5A – B = 5(a_x + 3a_z) – 5a_x – 2a_y + 6a_z = -2a_y + 21a_z.

c) Component of A along a_x, a_y and a_z are 1, 0 and 3 respectively.

d) 3A + B = 3a_x + 9a_z + 5a_x + 2a_y -6a_z = 8a_x + 2a_y +3a_z

a_{3A + B} = 3A + B/ |3A + B| = (8a_x + 2a_y + 3a_z)/ (77)^1/2

Q.2 Given Points P (1, -3, 5), Q (2, 4, 6) and R (0, 3, 8) find
a) Position vectors of P and R.
b) Distance vector r_QR.
c) Distance between Q and R.

Ans:

a) r_p = a_x – 3a_y + 5a_z.
r_R = 3a_y +8a_z.

b) r_QR = r_R -r_Q
= 3a_y +8a_z – (2a_x + 4a_y + 6a_z)
= - 2a_x – a_y + 2a_z.

c) |r_QR| = (2² + 1² + 2²)^1/2 = 3.

Q. 3 Find the unit vector along the line joining point (2, 4, 4) to point (-3, 2, 2)?

Ans:

Ā = (-3, 2, 2) – (2, 4, 4)
= (-5, -2, -2)

= - 0.87a_x - 0.35a_y - 0.35a_z

Q. 4 Given that A = 3a_x + 5a_y – 7a_z and B = a_x – 2a_y + a_z ; find

a) | 2B + 0.4A |
b) A.B - | B |²
c) A x B
Ans:

a) 2B + 0.4A
= 2a_x – 4a_y +2a_z + 0.4 (3a_x + 5a_y – 7a_z)
= 3.2a_x + 6a_y - 0.8a_z

| 2B + 0.4A | = (3.2² + 6² + 0.8²)^1/2 = 6.846

b) A.B - | B |²
= (3a_x + 5a_y -7a_z) . (a_x -2a_y + a_z) – (1² + 2² +1²)^1/2
= - 14 – (6)^1/2
= - 16.4494

c) A x B

Q.5 Given Vectors T = 2a_x – 6a_y + 3a_z and S = a_x + 2a_y +az; find

a) the scalar projection of T on S.
b) the vector projection of S on T.
c) the smaller angle between T and S.
Ans:

a)

= -0.286 a_x + 0.857a_y – 0.43a_z

c)

Sin θ_TS = 0.9129 => θ_TS = 65.91^o

Q.6 Let E = 3a_y + 4a_z and F = 4a_x – 10a_y + 5a_z

a) Find the component of E along F.
b) Determine a Unit vector perpendicular to both E and F.
Ans:

a)

= - 0.28a_x +0.71a_y - 0.35a_z

b)

= ± (0.94, 0.27, -0.21)

Q.7 E and F are vector fields given by E = 2xa_x + a_y +yza_z. and
F = xya_x – y2a_y +xyza_z. Determine:

a) | E| at (1, 2, 3)
b) Component of E along F at (1, 2, 3)
c) A vector perpendicular to both E and F at (0,1, -3) whose magnitude is unity?
Ans:

a) At (1, 2, 3), E = (2, 1, 6)

b) At (1, 2, 3), F = (2, -4, 6)

= 1.29 a_x – 2.57 a_y + 3.86 a_z

c) At (0, 1, -3), E = (0, 1, -3) and F = (0, -1, 0)

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Field Theory - Vector Multiplication - Dot and Cross product...

2009-12-06T16:24:00.012+05:30

Vector Multiplication:

When two vectors are multiplied the result is either a scalar or a vector depending on how they are multiplied.

The two important types of vector multiplication are:

- Dot Product/Scalar Product (A.B)
- Cross product (A x B)

DOT PRODUCT (A . B):

- Dot product of two vectors A and B is given as:

A . B = |A| |B| cosθ_AB

Where θ_AB is the angle formed between A and B.
Also θ ranges from 0 to π i.e. 0 ≤ θ_AB ≤ π

- The result of A.B is a scalar, hence dot product is also known as Scalar Product.

- If A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z) then A.B = A_xB_x + A_yB_y + A_zB_z

- If A.B = |A| |B|, then obviously cosθ_AB =1 which means θ_AB = 0^o
This shows that A and B are in the same direction or we can also say that A and B are parallel to each other.

- If A.B = - |A| |B|, then obviously cosθ_AB = -1 which means θ_AB = 180^o.
This shows that A and B are in the opposite direction or we can also say that A and B are antiparallel to each other.

- Similarly if A.B = 0, then cosθ_AB =0 which means θ_AB =90^o.
This shows that A and B are orthogonal or perpendicular to each other.

- Since we know the Cartesian base vectors are mutually perpendicular to each other, we have

a_x . a_x = a_y . a_y = a_z . a_z = 1

a_x . a_y = a_y . a_z = a_z . a_x = 0

CROSS PRODUCT ( A x B):

- Cross Product of two vectors A and B is given as:

A x B = |A| |B| sinθ_AB a_n

Where θ_AB is the angle formed between A and B and a_n is a unit vector normal to both A and B.
Also θ ranges from 0 to π i.e. 0 ≤ θ_AB ≤ π

- The cross product is an operation between two vectors and the output is also a vector.
If A = (A_x, A_y, A_z) and B = (B_x, B_y, B_z) then,

- The resultant vector is always normal to both the vector A and B.

- If A x B = 0, then sin θ_AB = 0 which means θ_AB = 0^o or 180^o;
This shows that A and B are either parallel or antiparallel to each other.

-Since we know the Cartesian base vectors are mutually perpendicular to each other, we have

a_x x a_x = a_y x a_y = a_z x a_z =0

a_x x a_y = a_z

a_y x a_z = a_x

a_z x a_x = a_y

SCALAR TRIPLE PRODUCT:

- A . (B x C) = B . (C x A) = C . (A x B)

- Volume of a parallelogram having A, B and C as edges is given by the Scalar Triple Product.

VECTOR TRIPLE PRODUCT:

- A x (B x C) = B (A . C) - C (A .B)

COMPONENT OF A VECTOR:

- Scalar Component A_B of vector A along vector B

A_B = Acosθ_AB = A |a_B|cosθ_AB = A.a_B

- Vector Component A_B of vector A along vector B
is the scalar product multiplied by the unit vector along B
i.e. A_B = (A.a_B) a_B

To Prove : (A * B) . A = 0.

A * B = a_x ( A_y B_z - A_z B_y ) - a_y ( A_x B_z - A_z B_x ) + a_z ( A_x B_y - A_y B_x )

A = A_x a_x + A_y a_y + A_z a_z

(A * B ) . A = A_x A_y B_z - A_x A_z B_y - A_x A_y B_z + A_y A_z B_x + A_x A_z B_y - A_y A_z B_x = 0.

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Field Theory - Vector Algebra - An Introduction...

2009-12-06T15:44:00.010+05:30

Most of the physical quantities are either scalar or vector quantities.

SCALAR QUANTITY:

- Scalar is a number that defines magnitude. Hence a scalar quantity is defined as a quantity that has magnitude only.

- A scalar quantity does not point to any direction i.e. a scalar quantity has no directional component.
For example when we say, the temperature of the room is 30^o C, we don’t specify the direction.

- Hence examples of scalar quantities are mass, temperature, volume, speed etc.

- A scalar quantity is represented simply by a letter – A, B, T, V, S.

VECTOR QUANTITY:

- A Vector has both a magnitude and a direction. Hence a vector quantity is a quantity that has both magnitude and direction.

- Examples of vector quantities are force, displacement, velocity, etc.

- A vector quantity is represented by a letter with an arrow over it.

UNIT VECTORS (a_A):

- When a simple vector is divided by its own magnitude, a new vector is created known as the unit vector
Mathematically, a_A = A / |A|

- A unit vector has a magnitude of one. Hence the name “unit vector”.

- A unit vector is always used to describe the direction of respective vector.
Rearranging the terms, we have

- Hence any vector can be written as the product of its magnitude and its unit vector.

- Unit Vectors along the co-ordinate directions are referred to as the base vectors.
For example unit vectors along X, Y and Z directions are a_x, a_y and a_z respectively.

Position Vector / Radius Vector ( r ):

- A Position Vector (r_Q)/ Radius vector defines the position of a point in space relative to the origin.
r_Q = xa_x + ya_y +za_z

- If the coordinates of some point is given as x =1, y =2 and z =3, then the position vector is defined as
r = a_x + 2a_y +3a_z.

- Hence Position vector is another way to denote a point in space.

- Position vector for Cartesian system in general is written as
r = xa_x + ya_y +za_z

But we cannot say the position vector for cylindrical and spherical coordinate system to be

r = ρa_ρ + φa_φ + za_z

r = ra_r +θa_θ + φa_φ
because θ and φ are not a unit of distance.

- Hence the correct position vector for cylindrical and spherical system is given as:

r = ρcosφa_x + ρsinφa_y + za_z

r = rsinθcosφa_x +rsinθsinφa_y + rcosθa_z

- A position vector should always be expressed using Cartesian base vectors (a_x, a_y, a_z).

- Displacement Vector is the displacement or the shortest distance from one point to another.

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