Dipole in an External Field

A dipole in a uniform external field experiences no net force but it does feel a torque that tries to align it with the field. This is why polar molecules orient in capacitor gaps — the basis of polarization and dielectrics.

Concept

Place a dipole of moment $\vec p$ in a uniform field $\vec E$. The $+q$ end is pushed by $q\vec E$, the $-q$ end by $-q\vec E$. The two forces are equal and opposite — net force is zero — but they form a couple producing torque.

Torque magnitude: $\tau = pE\sin\theta,$ where $\theta$ is the angle between $\vec p$ and $\vec E$.

Vector form: $\vec\tau = \vec p \times \vec E.$

Potential energy: $U = -\vec p \cdot \vec E = -pE\cos\theta.$

• $\theta = 0$: $U_\text{min} = -pE$ (stable equilibrium, dipole aligned with field).
• $\theta = \pi$: $U_\text{max} = +pE$ (unstable, anti-aligned).
• $\theta = \pi/2$: $U = 0$ (perpendicular).

Derivation

Torque. Force on $+q$ is $\vec F_+ = q\vec E$; on $-q$ is $\vec F_- = -q\vec E$. The arm from $-q$ to $+q$ is $\vec{2a}$. Torque about any point (here, the centre): $\vec\tau = \vec a \times \vec F_+ + (-\vec a)\times \vec F_- = 2\vec a \times q\vec E = (q\,\vec{2a})\times\vec E = \vec p \times \vec E.$ Magnitude: $\tau = pE\sin\theta$.

Potential energy. Work done by an external agent to rotate the dipole from $\theta_0$ to $\theta$ against the field is $W = \int_{\theta_0}^{\theta} \tau\,d\theta = \int_{\theta_0}^{\theta} pE\sin\theta\,d\theta = pE(\cos\theta_0 - \cos\theta).$ Taking $\theta_0 = \pi/2$ (where $U = 0$ by convention): $U(\theta) = -pE\cos\theta = -\vec p \cdot \vec E.$

Worked Example

A dipole of moment $p = 4\times 10^{-9}\,C\cdot m$ is placed in a field $E = 10^5\,N/C$ at $\theta = 60^\circ$.

Torque: $\tau = pE\sin\theta = (4\times 10^{-9})(10^5)\sin 60^\circ = 4\times 10^{-4}\times 0.866 = 3.46\times 10^{-4}\,N\cdot m$.

Potential energy: $U = -pE\cos 60^\circ = -(4\times 10^{-9})(10^5)(0.5) = -2\times 10^{-4}\,J$.

Work to rotate from $60^\circ$ to $180^\circ$: $W = U(\pi) - U(\pi/3) = +pE - (-pE/2) = \frac{3}{2}pE = 6\times 10^{-4}\,J.$

Common Confusions

• In a uniform field there is NO net force on a dipole. Only a torque.
• In a NON-uniform field, there IS a net force. The dipole gets pulled toward stronger field regions.
• $\theta = 0$ is the minimum of $U$, not a maximum. Stable equilibrium aligns $\vec p$ with $\vec E$.
• Sign of $U$: don't drop the minus. $U = -\vec p \cdot \vec E$.

Key Takeaways

• $\vec\tau = \vec p \times \vec E$, $\tau = pE\sin\theta$.
• $U = -\vec p \cdot \vec E = -pE\cos\theta$.
• Stable equilibrium: $\theta = 0$. Unstable: $\theta = \pi$.
• Net force = 0 in uniform field; nonzero in non-uniform field.
• Work to rotate $\theta_1 \to \theta_2$ is $pE(\cos\theta_1 - \cos\theta_2)$.