Potential due to Electric Dipole — Core Principles

Core Principles

An electric dipole consists of two equal and opposite point charges, $+q$ and $-q$, separated by a small distance $2a$. The electric dipole moment, $\vec{p}$, is a vector from $-q$ to $+q$ with magnitude $p = q(2a)$.

The electric potential at a point due to a dipole is the scalar sum of potentials from its constituent charges. For points far from the dipole ($r \gg a$), the potential $V$ at a distance $r$ from the center and at an angle $\theta$ with the dipole axis is given by $V = \frac{p \cos\theta}{4\pi\epsilon_0 r^2}$.

This formula shows a characteristic $1/r^2$ dependence, which is faster than the $1/r$ dependence for a single point charge. On the axial line ($\theta = 0^circ$ or $180^circ$), the potential is $V = \pm \frac{p}{4\pi\epsilon_0 r^2}$.

Crucially, on the equatorial line ($\theta = 90^circ$), the potential is always zero, although the electric field is not. This angular dependence is a key feature distinguishing dipole potential from point charge potential.

Important Differences

vs Potential due to a Point Charge

Aspect	This Topic	Potential due to a Point Charge
Source	Single isolated charge	Pair of equal and opposite charges (dipole)
Dependence on Distance (r)	$V \propto 1/r$	$V \propto 1/r^2$ (for $r \gg a$)
Dependence on Angle ($\theta$)	No angular dependence (spherically symmetric)	Depends on $\cos\theta$ (anisotropic)
Potential on Perpendicular Bisector	Non-zero (unless $r \to \infty$)	Zero (on the equatorial line)
Nature	Simpler, fundamental field	More complex, resulting from two point charges

The electric potential due to a point charge is spherically symmetric and decreases as $1/r$. In contrast, the potential due to an electric dipole is anisotropic, meaning it depends on both distance and angle. For distances much larger than the dipole's size, it decreases more rapidly, as $1/r^2$. A key distinction is that the potential is zero everywhere on the equatorial plane of a dipole, whereas a single point charge always produces a non-zero potential (except at infinity). These differences highlight the distinct spatial characteristics of their respective electric fields.