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


- 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:

 WA→B = - ∫AB (F . dl) = - Q ∫AB(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(VAB).

VAB = WA→B / Q 

= - ∫AB(E . dl) 

= - ∫InitialFinal (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:


VAB = - ∫AB (E . dl ) 

= - ∫APo (E . dl) - ∫ PoB (E . dl)

= - ∫PoB (E . dl)  -  (- ∫APo(E . dl)

= VB – VA


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:

WA→B = - Q [VB – VA] = - Q ∫AB (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 = - ∫PoP (E . dl)


The reference point Po 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 ( rB = 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. rA → ∞)






- 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 ….Qn located at points with position vectors r1, r2, r3…..rn, 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 = ρldl    → Q = ∫L (ρldl) → (Line Charge)

dQ = ρsds   → Q = ∫S (ρsds) → (Surface Charge)

dQ = ρvdv    → Q = ∫V (ρvdv) → (Volume Charge)



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...



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