Understanding Electric Potential and Equipotentials (AP Physics 2: Unit 2)

Electric potential energy (U)

Energy stored in a system of charges due to their relative positions; the electric force can do work as charges move.

Work done by the electric field (W_field)

Work done by the electric force on a charge; related to potential energy by W_field = −ΔU.

Change in electric potential energy (ΔU)

Difference in potential energy between final and initial states: ΔU = Uf − Ui.

External work (W_ext)

Work done by an external agent when moving a charge slowly (constant speed); equals the change in potential energy: W_ext = ΔU.

Energy approach (electrostatics)

Problem-solving method using work and potential energy (scalars) instead of force/field vectors; often avoids vector components.

Coulomb potential energy for two point charges

Potential energy of two point charges separated by r (zero at infinity): U = k(qQ)/r.

Coulomb’s constant (k)

Constant in electrostatics: k = 8.99×10^9 N·m^2/C^2.

Reference point (zero potential energy at infinity)

Common convention for point charges where U = 0 (and V = 0) when separation r → ∞; makes some U values negative for attraction.

Sign of potential energy for like charges

If qQ > 0 (like charges), then U > 0; positive external work is required to push them closer.

Sign of potential energy for opposite charges

If qQ < 0 (opposite charges), then U < 0; the system releases energy as charges attract.

Superposition of potential energy (multiple charges)

Total potential energy is the scalar sum over all distinct pairs: Utotal = Σ{i<j} k(qi qj)/r_ij (count each pair once).

Electric potential (V)

Electric potential energy per unit charge at a point: V = U/q.

Potential difference (ΔV)

Difference in electric potential between two points; relates to energy by ΔV = ΔU/q.

Volt (V) unit meaning

1 volt equals 1 joule per coulomb: 1 V = 1 J/C.

Energy–voltage relation

Change in potential energy when a charge q moves through ΔV: ΔU = qΔV.

Work–voltage relation (field)

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

Importance of the sign of charge (q)

In ΔU = qΔV and W_field = −qΔV, a negative q reverses the sign of energy/work compared with a positive charge.

Electric potential due to a point charge

Potential at distance r from source charge Q (zero at infinity): V = kQ/r.

Test charge independence of potential

Electric potential V depends on the source charges and position, not on the test charge; potential energy is U = qV.

Superposition of electric potential

Net potential is the algebraic (signed) sum of contributions: Vnet = Σ k(Qi)/r_i.

Electric field–potential connection (1D)

Electric field relates to spatial change in potential: E = −ΔV/Δx (field points toward decreasing V).

Uniform electric field potential change

For a uniform field, potential changes linearly with distance; magnitude relation: |ΔV| = Ed (sign depends on direction vs. E).

Equipotential line/surface

Set of points with the same electric potential; moving along it gives ΔV = 0 and thus ΔU = qΔV = 0.

Perpendicularity of E to equipotentials

Electric field is always perpendicular to equipotential lines/surfaces; otherwise it would do work along the equipotential, contradicting ΔV = 0.

Equipotential spacing and field strength

Closer equipotential lines/surfaces indicate a larger |E| because the potential changes more rapidly with distance (larger |ΔV| over smaller Δx).