Formulas/physics/Electrostatic Potential/PE of Charge in External Potential

PE of Charge in External Potential

Potential energy of a charge q placed at a point where the external potential is V. This energy is work done by external agent in bringing q from infinity.

Class 11Class 12

Derivation

An external potential $V$ exists at a point P (created by some charge distribution elsewhere). We bring charge $q$ from infinity to P.

Work done by the external agent:

W_{ext} = q(V_P - V_\infty) = q(V - 0) = qV

This work is stored as potential energy:

\boxed{U = qV}

Distinction from self-energy

This $U$ is the interaction energy between $q$ and the external distribution — not the self-energy of $q$ . We are treating $q$ as a test charge placed in a pre-existing potential.

Sign

$q > 0$ , $V > 0$ : $U > 0$ — energy stored; charge would accelerate toward lower $V$
$q < 0$ , $V > 0$ : $U < 0$ — bound configuration; work needed to remove the charge

Relation to force

\vec{F} = -\nabla U = -q\nabla V = q\vec{E}

Recovers the definition of the field as force per unit charge.

Remember

$U = qV$ is the potential energy of a single charge in an external potential. $U = q_1q_2/4\pi\varepsilon_0 r$ is the mutual PE of a pair. These are different quantities — do not confuse them.

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