Class 12 JEE Electric Potential and Electric Dipole Notes by Physics Wallah

Electric potential and electric dipoles connect electrostatic force, work, energy and electric fields. JEE questions often involve potential differences, charge interactions, dipole orientations and conductor properties, where incorrect signs, reference points or vector directions can lead to errors.

For focused revision, Physics Wallah’s Electric Potential and Electric Dipole notes for JEE Physics help you revisit essential concepts, formulae and their applications. Review the key results below and practise applying them to numerical problems to improve accuracy.

Electric Potential Energy

Conservative Forces and Potential Energy

A conservative force does work that depends only on the initial and final positions, not on the path followed. Electrostatic force is conservative, so its potential energy can be defined.

The work done by a conservative force is related to the change in potential energy by:

W_conservative = −ΔU

Potential energy belongs to a system of interacting objects rather than to an isolated charge. It represents the energy associated with their relative positions and arrangement.

Only changes in potential energy are physically measurable. Its numerical value depends on the reference point chosen.

Reference Point at Infinity

In electrostatics, infinity is generally chosen as the zero-reference point for isolated charge distributions:

U(∞) = 0

The potential energy at a point is therefore:

U(r) = −W_electric field (∞ → r)

If a system is assembled slowly from infinity, the external work done equals the increase in potential energy:

W_external = ΔU

When the initial potential energy is zero:

U = −W_electric field

U = W_external

Potential Energy of Two Point Charges

For two point charges q₁ and q₂, separated by a distance r, the electrostatic potential energy is:

U = k q₁ q₂ / r

where:

k = 1 / (4π ε₀)

The algebraic signs of both charges must be included.

• Same signs: U > 0, which means repulsion.

• Opposite signs: U < 0, which means attraction.

Potential energy is a scalar quantity, and its SI unit is the joule (J).

For a one-dimensional conservative interaction, the force is related to potential energy by:

F_r = −dU/dr

The negative sign indicates that the force acts towards decreasing potential energy.

Potential Energy of a System of Charges

For a system containing n point charges, the total electrostatic potential energy is the sum of the interaction energies of all distinct pairs:

U = Σ (i = 1 to n−1) Σ (j = i+1 to n) k q_i q_j / r_ij

Here, r_ij is the distance between charges q_i and q_j.

Each pair must be counted exactly once. For three charges, for example:

U = k [ q₁q₂/r₁₂ + q₂q₃/r₂₃ + q₁q₃/r₁₃ ]

Revision point: Calculate the potential energy of every distinct pair and add the signed values without double-counting.

Electric Potential

Electric potential at a point is the electrostatic potential energy per unit positive test charge:

V = U / q

Therefore:

U = qV

Electric potential is a scalar quantity. Its SI unit is the volt (V):

1 V = 1 J/C

The potential difference between two points is related to the work done by the electric field:

V_B − V_A = −∫ (A to B) E · dr

For a charge q moving between points A and B:

ΔU = qΔV

W_electric field = −qΔV

For slow movement without a change in kinetic energy:

W_external = qΔV

Always calculate potential difference as:

ΔV = V_final − V_initial

Potential Due to a Point Charge

The electric potential at a distance r from a point charge Q, taking zero potential at infinity, is:

V = kQ / r

The sign of the potential depends on the sign of the source charge:

• A positive charge produces positive potential.

• A negative charge produces negative potential.

For multiple point charges, the net potential is the algebraic sum of their individual potentials:

V = Σ k Q_i / R_i

Here, R_i is the distance from the i-th charge to the point where potential is calculated.

Important: Electric potential is a scalar, so its contributions are added algebraically. Electric field is a vector and must be added using vector rules.

Potential Due to Continuous Charge Distributions

For a continuous charge distribution, consider a small charge element dq at a distance r from the observation point.

The potential due to this element is:

dV = k dq / r

The total potential is obtained by integration:

V = ∫ k dq / r

Depending on the distribution, the charge element can be expressed as:

dq = λ dx

for a line charge, or

dq = σ dA

for a surface charge, where λ is linear charge density and σ is surface charge density.

Potential Due to a Charged Ring

For a uniformly charged ring of radius R and total charge Q, the potential at its centre is:

V_centre = kQ / R

At a point on the axis of the ring, at a distance x from its centre:

V = kQ / √(R² + x²)

Every charge element on the ring is at the same distance from an axial point, which simplifies the integration.

Potential Due to a Uniformly Charged Disc

Consider a disc of radius R, with uniform surface charge density σ. Divide the disc into concentric rings of radius y and thickness dy.

The charge on a ring element is:

dq = σ (2π y dy)

At an axial point a distance x from the disc, the potential contribution is:

dV = k dq / √(x² + y²)

Substituting the charge element and integrating over the disc gives:

V = k ∫ (0 to R) σ 2π y dy / √(x² + y²)

For x ≥ 0, taking zero potential at infinity:

V = 2π k σ [ √(x² + R²) − x ]

Using k = 1/(4π ε₀), this becomes:

V = (σ / 2ε₀) [ √(x² + R²) − x ]

The integration method is useful for determining the potential due to other continuous charge distributions.

Relation Between Electric Field and Potential

Electric field and electric potential are connected by the negative gradient of potential:

E = −∇V

In Cartesian coordinates:

E = −(∂V/∂x) i − (∂V/∂y) j − (∂V/∂z) k

For potential varying only along the x-direction:

E = −(dV/dx) i

The negative sign indicates that the electric field points towards decreasing potential.

For a uniform electric field and displacement d along the field:

ΔV = −Ed

For displacement perpendicular to a uniform electric field:

ΔV = 0

Thus, movement perpendicular to the field does not change the electric potential in a uniform electric field.

Equipotential Surfaces

An equipotential surface is a surface on which the electric potential is the same at every point.

Its important properties are:

1. Equipotential surfaces never intersect because a point cannot have two different electric potentials.

2. The electric field is perpendicular to an equipotential surface.

3. Work done by the electric field in moving a charge along an equipotential surface is zero.

4. The electric field has no tangential component along an equipotential surface.

The relation between work and potential difference is:

W_electric field = −qΔV

Since ΔV = 0 along an equipotential surface:

W_electric field = 0

Typical equipotential surfaces include:

• A point charge has concentric spheres as equipotential surfaces.

• An infinite uniform electric field has parallel planes.

• An infinite line charge has coaxial cylindrical surfaces.

JEE revision point: Equipotential surfaces are always perpendicular to electric field lines wherever the field is non-zero.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a finite distance.

For charges +q and −q, separated by distance d, the dipole moment magnitude is:

p = qd

The electric dipole moment is a vector directed from the negative charge towards the positive charge:

p: from −q to +q

Its SI unit is coulomb-metre (C m).

For an ideal dipole, or at observation distances much larger than the charge separation, the dipole approximation gives the standard field and potential expressions used in JEE Physics.

Torque on an Electric Dipole

When a dipole is placed in a uniform electric field, the forces on its two charges form a couple. The net force is zero, but the net torque may be non-zero.

The torque vector is:

τ = p × E

Its magnitude is:

τ = pE sinθ

where θ is the angle between the dipole moment and the electric field.

Torque is maximum when:

θ = 90°, τ_max = pE

Torque is zero when the dipole is parallel or antiparallel to the field.

Potential Energy of an Electric Dipole

The potential energy of a dipole in a uniform electric field is:

U = −p · E

Therefore:

U = −pE cosθ

The energy depends on the orientation of the dipole relative to the field.

• At θ = 0°, the torque is 0, the potential energy is −pE (minimum), and the equilibrium is stable.

• At θ = 90°, the torque is pE (maximum), the potential energy is 0, and the dipole is not in equilibrium.

• At θ = 180°, the torque is 0, the potential energy is +pE (maximum), and the equilibrium is unstable.

The dipole is in stable equilibrium when its moment is parallel to the field and in unstable equilibrium when its moment is antiparallel to the field.

Electric Field Due to an Electric Dipole

For an ideal dipole with moment p, at a distance r from its centre, the electric field depends on the position of the observation point.

Axial Field

On the axial line, the electric field magnitude is:

E_axial = 2kp / r³

Its direction is along the dipole moment on the positive axial side.

Equatorial Field

On the equatorial line, the electric field magnitude is:

E_equatorial = kp / r³

Its direction is opposite to the dipole moment.

Therefore:

E_axial = 2 E_equatorial

These expressions apply to an ideal dipole or to points sufficiently far from a finite dipole compared with its charge separation.

Electric Field at a General Point

For a point at distance r from the dipole centre, making an angle θ with the dipole moment, the field magnitude in the dipole approximation is:

E = (kp / r³) √(1 + 3cos²θ)

This expression gives the axial and equatorial results when θ = 0° and θ = 90°, respectively.

Electrostatic Pressure and Energy Density

Electrostatic Pressure

A charged conductor experiences an outward electrostatic pressure due to the repulsion between its surface charges.

The pressure is:

P = σ² / (2ε₀)

Using the electric field immediately outside the conductor,

E = σ / ε₀

the pressure can also be expressed as:

P = ½ ε₀ E²

Electrostatic Energy Density

The energy stored per unit volume in an electric field in vacuum is:

u = ½ ε₀ E²

The SI unit of energy density is joule per cubic metre (J/m³), whereas the SI unit of pressure is pascal (Pa).

Although the two expressions have the same mathematical form in this context, pressure and energy density represent different physical quantities.

The Electric Potential and Electric Dipole Class 12 JEE Notes by PW cover essential concepts, including electrostatic potential energy, potential difference, equipotential surfaces, electric dipole fields, conductor properties and electrostatic shielding. Revise the key formulas, sign conventions and field-potential relationships to strengthen your conceptual understanding and improve accuracy when solving JEE Physics numerical and conceptual questions.