Potential Energy of a Dipole in an External Field

Potential Energy of a Dipole in an External Field

When an electric dipole (charges $+q$ and $-q$ separated by distance $2a$) is placed in a uniform external electric field $\mathbf{E}$, it experiences no net force but a torque $\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}$. This torque tends to rotate the dipole to align with the field. To change its orientation, an external torque must do work against this torque. This work is stored as potential energy of the dipole in the field.

Work Done by External Torque

• Let the dipole be rotated from an initial angle $\theta_0$ to a final angle $\theta_1$ (measured from the direction of $\mathbf{E}$) at an infinitesimal speed (no angular acceleration).
• The external torque exactly cancels the electric torque at every step.
• The work done by the external torque is:

This work is stored as the change in potential energy of the system.

Defining Potential Energy $U(\theta)$

• We can associate a potential energy $U(\theta)$ with the dipole's orientation.
• There is a freedom to choose a reference angle where $U = 0$. A natural choice is $\theta_0 = \pi/2$ (dipole perpendicular to the field). Why? At $\theta = \pi/2$, the work done in bringing $+q$ and $-q$ from infinity to their positions against the field cancels out, making the constant term zero.
• With this choice, the potential energy at any angle $\theta$ is:

• $p$: magnitude of dipole moment ($p = q \times 2a$)
• $E$: magnitude of uniform electric field
• $\theta$: angle between $\mathbf{p}$ and $\mathbf{E}$

Alternative Derivation Using Potential Energy of Two Charges

• For two charges $+q$ and $-q$ at positions $\mathbf{r}_1$ and $\mathbf{r}_2$, the potential energy in an external field is:

where $V(\mathbf{r})$ is the electrostatic potential due to the external field.

• For a uniform field $\mathbf{E}$, the potential difference between the two charges is:

• Substituting: $U' = -q(2a)E \cos \theta - \frac{q^2}{4\pi\varepsilon_0 (2a)} = -\mathbf{p} \cdot \mathbf{E} - \text{constant}$ …