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


- Consider an infinite line charge carrying a charge per unit length of ρL 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 ∫0l  (ρ dφ dz) aρ → S = 2πρl aρ


Hence applying gauss law, we have


→ Qenc = Dρ 2πρl = ρL l



INFINITE SHEET OF CHARGE


- Consider an infinite sheet charge carrying a charge per unit surface of ρs 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.


ALSO READ: 

- 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.

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



Your suggestions and comments are welcome in this section. If you want to share something or if you have some stuff of your own, please do post them in the comments section.

Comments

Post a Comment

Popular posts from this blog

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

Image
A Vector in Cylindrical system is represented as (A ρ , A φ , A z ) or A = A ρ a ρ + A φ a φ + A z a z Where a ρ , a φ and a z are the unit vectors in ρ, φ and z direction respectively. The physical significance of each parameter of cylindrical coordinates:      - The value ρ indicates the distance of the point from the z-axis. It is the radius of the cylinder .      - The value φ, also called the azimuthal angle , indicates the rotation angle around the z-axis. It is basically measured from the x axis in the x-y plane. It is measured anti-clockwise.      - The value z indicates the distance of the point from z-axis. It is the same as in the Cartesian system. In short, it is the height of the cylinder. Range of the variables: It defines the minimum and the maximum value that ρ, φ and z can have in Cartesian system.  0 ≤ ρ ≤ ∞  0 ≤ φ ≤ 2π - ∞ ≤ z ≤ ∞   Figure shows Point P and Unit vectors in Cylindrical Co-ordinate System. Cylindrical System - Unit v
Read More »

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

Image
- 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 ) R 2 = a 2 + h 2 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 / ( a 2 + h 2 ) 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
Read More »

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

Image
- Consider a circular disc of radius ‘a’ which carries a uniform surface charge density ρ s , C /m 2 . - 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 = (ρ 2 + h 2 ) 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: ALSO READ:  - Gauss's Law - Theory. - Gauss's Law - Application To a Point charge. - Gauss's Law - Application To An Infinite Line Charge. - Gauss's
Read More »