ELECTROSTATIC POTENTIAL AND CAPACITANCE

Electric Potential

Electric Potential at a point in an electric field is the amount of work done in moving a
positive charge of 1 coulomb from infinity to that point against the electrostatic forces.

Work done
Electrostatic Potential =
Charge

Potential Difference

The potential difference between two points in an electric field may be defined as the amount
of work done in moving a unit positive charge from one point to the other against the
electrostatic forces.

SI unit of potential difference is volt (V)

1 Joule
1 Volt =
1 Coulomb
1V =1 JC-1

The potential difference between two points in an electric field is said to be 1 volt if 1 joule
of work has to be done in a moving a positive charge of 1 coulomb from one point to the
other against the electrostatic forces.

Properties of Electrostatic Potential Difference:

1. Scalar Quantity: The potential difference is a scalar quantity and does not have direction,
only magnitude.

2. Independent of Path: In an electrostatic field, the potential difference between two points
depends only on the position of the points and not on the path taken by the charge.

3. Work Done: It measures the work required to move a unit positive charge between two
points. A positive potential difference implies work is done against the field, while a negative
potential difference implies the field does work.

4. SI Unit: The SI unit of electrostatic potential difference is the volt (V) .

5. Relation to Electric Field: The potential difference between two points is related to the
electric field by the equation:
𝐵

∆𝑉 = − ∫ ⃗⃗⃗
𝐸. 𝑑𝑟
𝐴

where 𝐸⃗ is the electric field and 𝑑𝑟 is the displacement vector from along the path from A to
B
Electric Potential Due to a Point Charge:

The electric potential at a distance r from a point charge q is the amount of work done to
bring a unit positive charge from infinity to the point.

1 𝑞
𝑉=
4𝜋𝜀0 𝑟

Electric Potential Due to a Dipole

Due to a short electric dipole at a distance r from its centre

1 𝑝
i) at its axis is 𝑉 = 4𝜋𝜀 2
0 𝑟

ii) at its equitorial position V=0

iii) At a general point having polar coordinates(r,θ) with respect to the centre of the dipole is

1 𝑝𝑐𝑜𝑠𝜃
𝑉=
4𝜋𝜀0 𝑟 2

Electric potential of a dipole Electric potential of a single charge

1 It depends on r as well as angle It depends on r
between 𝑟 and 𝑝.
2 It has cylindrical symmetry It has spherical symmetry

3 1 1
At large distance it falls off as 𝑟 2. It falls off as 𝑟 .

Electric Potential Due to a System of Charges:

For a system of charges , the total electric potential at a point is the algebraic sum of
the potentials due to individual charges.
𝑛
1 𝑞𝑖
𝑉= ∑
4𝜋𝜀0 𝑟𝑖
𝑖=1

Key Points:

- Point Charge: Potential varies inversely with distance r

- Dipole: Potential depends on both distance r and angle θ
- System of Charges: The potential is the sum of potential due to individual charges.
Equipotential Surface

An equipotential surface is a surface where the electric potential at every point is the same.
This means that if a charge is moved along this surface, no work is done by or against the
electric field because the potential difference between any two points on the surface is zero.

Properties of Equipotential Surfaces

1. No Work Done: Moving a charge along an equipotential surface requires no work, as the
potential difference is zero.

2. Perpendicular to Electric Field: Equipotential surfaces are always perpendicular to the

electric field lines. This is because work would otherwise be done if the charge moved in the
direction of the field.

3. Closer Surfaces, Stronger Field: In regions where the equipotential surfaces are closer
together, the electric field is stronger. This is because the electric potential changes more
rapidly over a small distance.

4. Shape Depends on Charge Distribution:

- For a point charge, the equipotential surfaces are concentric spheres around the charge.

- For a uniform electric field, the equipotential surfaces are parallel planes.

- For a dipole, the surfaces are more complex, but still symmetric.

Equipotential surfaces due to various charge systems

i) Positive point charge

 ii) Electric dipole

iii) Two equal positive charges

iv) Uniform electric field

Importance of Equipotential Surfaces

Like the lines of force, equipotential surfaces give a visual picture of both the direction and
the magnitude of field 𝐸⃗ in a region of space.
Relation between field and potential

𝑑𝑉
𝐸= −
𝑑𝑟
Electric field = Negative of potential gradient at any point

Properties relating electric field to electric potential

i) Electric field is in that direction in which the potential decreases steepest.

ii) The magnitude of the electric field is equal to the change in the magnitude of
potential per unit displacement (called potential gradient)

Electric potential energy

The electric potential energy of a system of point charges may be defined as the amount of
work done in assembling the charges at their locations by bringing them in, from infinity. It is
the potential energy possessed by a system of charges by virtue of their positions.

Potential energy of a system of two point charges

1 𝑞1 𝑞2
𝑈=
4𝜋𝜖0 𝑟12

Potential energy of a system of three point charges

1 𝑞1 𝑞2 𝑞1 𝑞3 𝑞2 𝑞3
𝑈= + +
4𝜋𝜖0 𝑟12 𝑟13 𝑟23

Potential energy of a system of N point charges

1 𝑞𝑖 𝑞𝑗
𝑈= ∑
4𝜋𝜖0 𝑟𝑖𝑗
𝑎𝑙𝑙 𝑝𝑎𝑖𝑟𝑠

Potential Energy in an External Field

The work done in bringing a charge q from infinity to a point P will be 𝑞𝑉.

𝑊 = 𝑞𝑉

This work done is stored as the potential energy of the charge q.

𝑈 = 𝑞𝑉

P.E of a charge in external field = Charge x External electric potential

 𝑈
As, 𝑉 = 𝑞

Electric potential at a given point in an electric field is the potential energy of a unit positive
charge at that point.

Potential Energy of a system of two charges in an External Electric field

Let 𝑉(𝑟⃗⃗⃗1 ) and 𝑉(𝑟⃗⃗⃗2 ) be the electric potentials of the field 𝐸⃗ at the points having position
vectors ⃗⃗⃗
𝑟1 𝑎𝑛𝑑 ⃗⃗⃗
𝑟2 .

Total potential energy of the system = Work done in assembling the two charges.

1 𝑞1 𝑞2
𝑈 = 𝑞1 𝑉(𝑟⃗⃗⃗1 ) + 𝑞2 𝑉(𝑟⃗⃗⃗2 ) + . 2
4𝜋𝜖0 𝑟12

Potential Energy of dipole placed in a Uniform electric field

A dipole of dipole moment 𝑝 is placed in an external field 𝐸⃗ making an angle 𝜃 with it, as
shown in figure. Two equal and opposite forces +𝑞𝐸⃗ and −𝑞𝐸⃗ act on its two ends.

Two forces form a couple. The torque exerted by the couple will be

τ = qE x 2asinθ = pEsinθ

If the dipole is rotated through a small angle 𝑑𝜃 against the torque acting on it. Then the
small work done will be

𝑑𝑊 = 𝜏𝑑𝜃 = 𝑝𝐸𝑠𝑖𝑛𝜃𝑑𝜃

The total work done in rotating the dipole from its orientation making an angle 𝜃1 , with the
direction of the field to 𝜃2 will be
θ2

W = ∫ dW = ∫ pEsinθ
θ1

= 𝑝𝐸(𝑐𝑜𝑠𝜃1 − 𝑐𝑜𝑠𝜃2 )
This work done is stored as the Potential Energy U of the dipole.

𝑈 = 𝑝𝐸(𝑐𝑜𝑠𝜃1 − 𝑐𝑜𝑠𝜃2 )

If initially the dipole is oriented perpendicular to the direction of the field (𝜃1 = 900 ) and then
brought to some orientation making an angle 𝜃 with the field (𝜃2 = 𝜃), then the potential
energy of the dipole will be

𝑈 = 𝑝𝐸(𝑐𝑜𝑠90𝑜 − 𝑐𝑜𝑠𝜃)

= 𝑝𝐸(0 − 𝑐𝑜𝑠𝜃)

Or 𝑈 = −𝑝𝐸 𝑐𝑜𝑠𝜃 = −𝑝 . 𝐸⃗