Electric Potential Due to an Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. While the net charge of a dipole is zero, it produces a non-zero electric potential at points in the surrounding space because the two charges are spatially separated and their individual contributions do not cancel exactly. The electric potential due to a dipole at any point depends on the distance of that point from the centre of the dipole and the angle the position vector makes with the dipole axis. This forms an important application of the superposition principle in electrostatics.

Setup and Notation of Electric Potential Due to an Electric Dipole

Consider an electric dipole consisting of charges −q and +q separated by a distance 2a, with the centre of the dipole at the origin. The dipole moment is:

p = q × 2a

directed from −q to +q along the dipole axis.

Let P be a point at distance r from the centre of the dipole, making an angle θ with the dipole axis.

Let r₁ = distance of P from −q Let r₂ = distance of P from +q

General Expression for Electric Potential

By the superposition principle, the total electric potential at P is:

V = V₊ + V₋ = kq/r₂ − kq/r₁

Where k = 1/4πε₀.

Using the geometry of the dipole and the approximation r >> a:

r₁ ≈ r + a cosθ

r₂ ≈ r − a cosθ

Substituting:

V = kq(1/r₂ − 1/r₁)

V = kq × [(r + a cosθ − r + a cosθ) / (r² − a²cos²θ)]

V = kq × 2a cosθ / (r² − a²cos²θ)

Since r >> a, the term a²cos²θ is negligible compared to r²:

V = kp cosθ / r²

Or equivalently:

V = p cosθ / 4πε₀r²

Where p = q × 2a is the magnitude of the dipole moment.

Potential at Special Points

On the Axial Line (θ = 0° or θ = 180°)

At θ = 0° (along the direction of dipole moment):

V = kp / r²

The potential is positive and maximum along this direction.

At θ = 180° (opposite to dipole moment):

V = −kp / r²

The potential is negative and equal in magnitude to the axial value.

On the Equatorial Line (θ = 90°)

V = kp cos90° / r² = 0

The electric potential is zero at all points on the equatorial plane of the dipole. This does not mean the electric field is zero on the equatorial line. It means only the potential is zero.

Key Observations in Electric Potential Due to an Electric Dipole

Dependence on Distance

The potential due to a dipole falls as 1/r², which is faster than the 1/r dependence of a point charge. This is because the dipole has zero net charge and the contributions of the two charges partially cancel, resulting in a weaker and faster-decaying potential.

Dependence on Angle

The potential depends on cosθ, making it directionally asymmetric. It is maximum on the axial line, zero on the equatorial line, and varies continuously between these extremes at intermediate angles.

Sign of potential:

For 0° ≤ θ < 90°, cosθ > 0 and V > 0.
For 90° < θ ≤ 180°, cosθ < 0 and V < 0.
At θ = 90°, V = 0.

Conclusion

The electric potential due to a dipole at a general point is V = kp cosθ / r², derived using the superposition principle and the approximation r >> a. On the axial line, the potential is ±kp/r² depending on direction. On the equatorial plane, the potential is zero at all points. The potential falls as 1/r², faster than a point charge, and is directionally dependent through the cosθ factor. These results are fundamental to understanding dipole behaviour in external fields and the electrostatics of polar molecules.