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 = ΔU.

Point-charge potential energy (two charges)

With U(∞)=0, two point charges have U(r) = k(q1 q2)/r.

Coulomb constant (k)

k = 1/(4πϵ0), the proportionality constant in Coulomb’s-law-based potential and potential energy formulas.

Sign of U for like charges

For q1q2 > 0, U > 0; energy must be added to bring like charges closer together.

Sign of U for opposite charges

For q1q2 < 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 Σ{i

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 = 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: ΔV = Vf − Vi.

Potential energy change from potential difference

For a charge q moving between two points: ΔU = qΔV (sign depends on q).

Work–potential relation

Work done by the electric field in moving charge q through ΔV: W_field = −qΔV.

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(∞)=0, a point charge Q produces V(r) = kQ/r; sign of V matches sign of Q.

Scalar superposition of potential

Because V is a scalar, potentials add by ordinary addition: Vtotal = k Σi Qi/ri.

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)

ΔV = Vb − Va = −∫_a^b E·dℓ (path-independent in electrostatics).

Electric field from potential (gradient)

Field is the negative gradient of potential: E = −∇V (in 1D: E_x = −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(dq)/r and integrate: V = k∫(dq/r), with dq expressed via λdl, σdA, or ρdτ depending on the distribution.