Potential Energy in External Field — Core Principles

Core Principles

Potential energy in an external electric field is the energy a charge or a system of charges possesses due to its position within a pre-existing electric field. For a single point charge $q$ at a location where the external potential is $V$, its potential energy is $U = qV$.

This represents the work an external agent must do to bring the charge from infinity to that point without acceleration. For a system of two charges $q_1$ and $q_2$ at $vec{r_1}$ and $vec{r_2}$ respectively, the total potential energy includes their individual potential energies in the external field and their mutual interaction energy: $U = q_1 V(vec{r_1}) + q_2 V(vec{r_2}) + \frac{1}{4piepsilon_0} \frac{q_1 q_2}{r_{12}}$.

For an electric dipole with moment $vec{p}$ in a uniform external electric field $vec{E}$, its potential energy is $U = -vec{p} cdot vec{E}$. This energy is minimum when the dipole aligns with the field ($heta=0^circ$) and maximum when it is anti-aligned ($heta=180^circ$).

Understanding these formulas and their sign conventions is crucial for NEET, as they govern the behavior of charges and dipoles in electric environments.

Important Differences

vs Potential Energy of a Single Charge vs. Potential Energy of an Electric Dipole in an External Field

Aspect	This Topic	Potential Energy of a Single Charge vs. Potential Energy of an Electric Dipole in an External Field
Nature of Entity	Single point charge ($q$)	Electric dipole ($vec{p}$)
Formula for Potential Energy	$U = qV(vec{r})$	$U = -vec{p} cdot vec{E}$ (for uniform field)
Dependence	Depends on the magnitude and sign of the charge, and the scalar electric potential at its location.	Depends on the magnitude of the dipole moment, the magnitude of the electric field, and the angle between them (orientation).
Equilibrium Conditions	A single charge in an external field does not have 'equilibrium' in the same sense as a dipole's orientation. It will move to minimize its potential energy (e.g., positive charge moves to lower potential).	Stable equilibrium at $ heta = 0^circ$ ($U = -pE$), unstable equilibrium at $ heta = 180^circ$ ($U = +pE$).
Force/Torque Experienced	Experiences an electric force $vec{F} = qvec{E}$.	Experiences a net force of zero in a uniform field, but experiences a torque $vec{ au} = vec{p} imes vec{E}$.

The potential energy of a single charge in an external field is a direct product of its charge and the potential at its location ($qV$), reflecting the work done to bring it there. In contrast, the potential energy of an electric dipole in a uniform external field is given by the negative dot product of its dipole moment and the electric field ($-vec{p} cdot vec{E}$), highlighting its dependence on orientation. A single charge experiences a force, while a dipole in a uniform field experiences a torque that tends to align it, leading to distinct stable and unstable equilibrium orientations based on its potential energy.