Short on time? This page is the recap. For explanations, Figures 1 and 2 and the worked examples, use the full study guide.
Recap
- Electric potential V is the electric potential energy per unit charge at a point. It is a property of the point, not of the test charge. V is zero infinitely far away.
- V is a scalar: for several point charges, add each kq/r with its sign.
- Potential difference ΔV between two points is the change in U_E per coulomb when a charge moves between them. Batteries keep a potential difference by separating charge chemically.
- A conductor in equilibrium is at one potential throughout. Conductors joined together share electrons until their surfaces are at the same potential, not until they hold equal charge.
- The average field between two points is |ΔV| divided by the distance along the field. E points toward decreasing V.
- Equipotential lines are perpendicular to the field. Close lines mean a strong field. Moving along an isoline, the field does no work.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| V = U_E / q | V in volts, 1 V = 1 J/C | link potential and potential energy |
| V = Σ kqᵢ / rᵢ | k = 9.0 × 10⁹ N·m²/C², q with sign, r in m | find V from up to four point charges |
| ΔU_E = qΔV | J; q with sign | find the energy change when a charge moves |
| |E| = |ΔV / Δr| | V/m = N/C; Δr along the field | find the average field and its size |
| Charge released at rest | Moves toward | Its U_E |
|---|---|---|
| Positive | lower V (along E) | decreases |
| Negative | higher V (against E) | decreases |
Assumptions behind the numbers
- V = 0 infinitely far from all charges.
- Charges are point charges, or spheres that act like point charges from outside.
- Conductors are in electrostatic equilibrium (no charge is still moving).
- |ΔV/Δr| gives an average field; it equals the field everywhere only where the field is uniform.
Mistakes to avoid
- Treating V as a vector. No components, no angles.
- Losing the sign of q in kq/r.
- Using r² in the potential. That is the field formula.
- “V = 0 so E = 0.” Midway between +q and −q, V = 0 but E ≠ 0.
- Field pointing uphill. E points toward lower V.
- “Equal charge after contact.” Equal potential, not equal charge.
Quick self-check
- What is the potential 0.30 m from a +2.0 nC point charge? (60 V)
- Two parallel equipotentials at 0 V and 12 V are 4.0 mm apart. What is the field between them, and which way does it point? (3.0 × 10³ V/m, from the 12 V line toward the 0 V line)
- How much work does the field do on a charge carried along an equipotential line? (Zero)
Next: practice questions.