Electric Potential in Electrostatics (AP Physics C: E&M Unit 1)

Electric potential energy (U)

Energy stored in a system due to the configuration (positions) of electric charges; not a property of an isolated single charge.

Conservative electric force (electrostatics)

Electric force is conservative for time-independent fields, so a potential energy function U can be defined and work depends only on initial and final positions.

Work done by the electric field

Relationship to potential energy change: W_field = −ΔU.

External work (quasistatic move)

If an external agent moves a charge slowly so kinetic energy doesn’t change, the external work equals the potential energy increase: $W_{ext} = \Delta U$.

Point-charge potential energy (two charges)

With $U(\infty)=0$, two point charges have $U(r) = k \frac{q_{1} q_{2}}{r}$.

Coulomb constant (k)

$k = \frac{1}{4\pi\epsilon_{0}}$, the proportionality constant in Coulomb’s-law-based potential and potential energy formulas.

Sign of U for like charges

For $q_{1}q_{2} > 0$, $U > 0$; energy must be added to bring like charges closer together.

Sign of U for opposite charges

For $q_{1}q_{2} < 0$, $U < 0$; energy is released as opposite charges move closer together.

Pairwise sum for potential energy (many charges)

Total potential energy for multiple point charges: $U = k \sum_{i<j} \frac{q_{i} q_{j}}{r_{ij}}$ (sum over distinct pairs to avoid double counting).

Potential-energy-from-potential formula

Equivalent system energy expression: U = (1/2) Σi qi Vi, where Vi is due to all other charges; 1/2 prevents double counting.

Electric potential (V)

Electric potential at a point is potential energy per unit charge: $V = \frac{U}{q}$ (a property of the field/source charges, not the test charge).

Volt (V)

Unit of electric potential: 1 V = 1 J/C.

Potential difference (ΔV)

Change in electric potential between two points: $\Delta V = V_{f} - V_{i}$.

Potential energy change from potential difference

For a charge $q$ moving between two points: $\Delta U = q\Delta V$ (sign depends on $q$).

Work–potential relation

Work done by the electric field in moving charge $q$ through $\Delta V$: $W_{field} = -q \Delta V$.

Reference potential

Potential is defined up to an additive constant; only potential differences are physically measurable (often choose V(∞)=0 when it converges).

Potential of a point charge

With $V(\infty)=0$, a point charge $Q$ produces $V(r) = \frac{kQ}{r}$; sign of $V$ matches sign of $Q$.

Scalar superposition of potential

Because $V$ is a scalar, potentials add by ordinary addition: $V_{total} = k \sum_{i} \frac{Q_{i}}{r_{i}}$.

Equipotential surface

A set of points with the same V; moving along it gives ΔV=0 ⇒ ΔU=0, so the electric field does no work along the equipotential.

Perpendicularity of E and equipotentials

Electric field lines cross equipotential surfaces at right angles because E points in the direction of greatest decrease of V.

Potential difference from the electric field (line integral)

$\Delta V = V_{b} - V_{a} = -\int_{a}^{b} E\cdot d\ell$ (path-independent in electrostatics).

Electric field from potential (gradient)

Field is the negative gradient of potential: $E = -\nabla V$ (in 1D: $E_{x} = -\frac{dV}{dx}$).

Conductor in electrostatic equilibrium

Inside conducting material, E=0 and the potential is constant throughout the conductor (equal to the surface potential).

Conducting sphere potential

For a conducting sphere (radius R, charge Q, V(∞)=0): V(r)=kQ/r for r≥R and V(r)=kQ/R for r≤R (constant inside).

Potential from charge distributions (integration)

Use $dV = k \frac{dq}{r}$ and integrate: $V = k \int \frac{dq}{r}$, with $dq$ expressed via $\lambda dl$, $\sigma dA$, or $\rho d\tau$ depending on the distribution.