# Physics | 12 | 2

## Unit — Electrostatics Chapter — 2| Electrostatic Potential and Capacitance

Electric Potential
Electric Potential at any point in an electric field is defined as the amount of work done in moving a unit positive charge from infinity to that point against the electrostatic forces.

Electric Potential (VP) = W/q₀
Note :
1) Here We have taken the test charge so small that it does not disturb the distribution of the source charge.
2) We just apply that much external force on the test charge that it just balances the repulsive electric force on it and hence does not produce any acceleration in it.
⚫ It is a Scalar quantity.
⚫ It’s SI unit is J/C or Volt (V).
⚫ Electric Potential at infinity as taken as equal to zero. (V∞=0)

Define 1 Volt?
The potential at a point is said to be 1 Volt if 1 Joule of work is done in moving a positive charge of 1 coulomb from infinity to that point against electrostatic forces.

Potential Difference
The potential difference between two points (A & B) in an electric field is defined as the amount of work done in moving a unit positive charge from one point to the other against electrostatic forces.

FIG

The potential difference between A and B is VB-VA = WAB/q₀

⚫ It is a Scalar quantity.
⚫ It’s SI unit is J/C or Volt (V).

Electric Potential due to a Point Charge

DERIVATION

W = kq q₀ /r
V = W/q₀ or kq/r
V = kq/r
Note :
1) Clearly
V ∝ 1/r but E ∝ 1/r2
Graphical Variation of E and V with r in the same graph.
2) Also
r → 0
V → ∞

Electric Potential at an axial point of a dipole

DERIVATION

V = p/4πε₀ (r2-a2)
For short dipole
V=p/4πε₀r2

Electric Potential at an equatorial point of a dipole

DERIVATION

V = 0

Electric Potential at any General Point due to a dipole
DERIVATION

V=kpcos θ/r2
Special Cases :
1) At axial point
θ = 0° or 180°
So V = ±kp/r2
2) At equatorial point
θ = 90°
So V=0

Questions :
Give an example of a point where the electric field is zero but the electric potential is not zero?
Give an example of a point where the electric potential is zero but the electric field is not zero?

Electric Potential due to a system of charges
We use superposition Principle to find the net potential due to a system of point charges at a point.

Electric Potential due to a uniformly charged thin spherical shell.
Electric Potential when the point P lies outside the sphere
V=kq/r2 (r>R)
Electric Potential when the point P lies on the sphere
V=kq/R2(r=R)
Electric Potential when the point P lies outside the sphere
V= kq/R2 (r<R)

Relation between Electric Field and Electric Potential
DERIVATION

E=-dV/dr

The quantity dV/dr is the rate of change of potential with the distance which is called Potential gradient.
Here -ve sign shows that the direction of the electric field is in the direction of decreasing potential.

Equipotential Surfaces and their Properties

Any surface that has same electric potential at every point on it is called an equipotential surface. For eg. The surface of a charged conductor is an equipotential surface.

Properties of Equipotential Surfaces :
⚫ No Work is done in moving a test charge over an equipotential surface.
Reason:
(1)
(2)

⚫ The electric field is always normal to the equipotential surface at every point.
Reason: If the electric field is not normal to the surface, then there must be some work done in displacing the test charge over the equipotential surface, which violates the first property.

⚫ Equipotential surfaces are closer together in the regions of strong electric field and farther apart in the region of a weak field.

We know
E = – dv/dr
or dr = -dv/E
when dv is constant
dr ∝ 1/E
Thus the spacing between the equipotential surfaces will be smaller in the regions where the electric field is stronger and vice versa.

⚫ No two equipotential surfaces can intersect each other.
Reason: If they intersect, then at the point of intersection, there would be two values of electric potential which is not possible.

Equipotential Surfaces of various charge systems
i) Equipotential surfaces of a positive point charges:
ii)Equipotential surfaces of two equal and opposite point charges: (Dipole)
iii) Equipotential surfaces of two equal positive charges.
iv) Equipotential surfaces for a uniform electric field.

Electric Potential Energy :

Effect of the dielectric when the battery is kept disconnected from the capacitor :

Charge
Q = Q₀
Electric Potential
V = V₀/ K
Electric Field
E=E₀/K
Capacitance
C = KC₀
Energy Stored
U = U₀/K

Effect of the dielectric when the battery is kept disconnected from the capacitor :

Charge
Q = KQ₀
Electric Potential
V = V₀
Electric Field
E=E₀
Capacitance
C = KC₀
Energy Stored
U = KU₀