Inside This Unit: The Full Breakdown
This unit develops electric potential and its relationship to the field through V = −∫E·dl and E = −dV/dr, and computes potential from charge distributions.
Why it matters
Potential is a scalar, often easier to compute than the field, and the field-potential calculus link is heavily tested.
Key concepts
- V = U/q; for a point charge V = kq/r (a scalar).
- Potential from a field: V = −∫E·dl; field from potential: E = −dV/dr.
- Potential of a distribution integrates scalar contributions ∫k dq/r.
- Equipotentials are perpendicular to field lines; no work is done along them.
Field and Potential
Potential difference is the negative line integral of the field, ΔV = −∫E·dl, and conversely the field is the negative gradient, E = −dV/dr. Because potential is a scalar, computing V from a distribution avoids the vector bookkeeping needed for the field.
Equipotentials and Energy
Equipotential surfaces are perpendicular to field lines; moving a charge along one does no work. Work to move a charge between points is W = qΔV, and closely spaced equipotentials indicate a strong field.
AP exam tip
If asked for the field given V(r), differentiate (E = −dV/dr); if asked for V given E(r), integrate (V = −∫E·dl). Mixing these up is a common error.
Connections to other units
- Unit 1: The potential is the integral of the field found by Gauss’s law.
- Unit 3: Capacitor voltage is a potential difference.
