Short on time? This page is the recap. For explanations, both figures and the worked examples, use the full study guide.
Recap
- Potential V belongs to a place (J/C = V). Potential energy U_E belongs to a system of charges (J). For a charge q at potential V, U_E = qV.
- Moving a charge between two points changes U_E by qΔV. Only the end points matter, not the path.
- If only electric forces do work, the total K + U_E of the charge–field system is constant.
- Released from rest, a positive charge moves toward lower V (with the field); a negative charge moves toward higher V (against the field). Both lose U_E and gain K.
- Along an equipotential line ΔV = 0, so U_E and K do not change.
- Gravity is negligible for electrons, protons and ions.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| ΔU_E = qΔV | q in C (with sign), ΔV = V_f − V_i in V | find the change in potential energy |
| ΔK + ΔU_E = 0 | all in J | link speed changes to potential differences |
| ΔK + ΔU_E = W_ext | W_ext = work by an outside force | include a hand, machine or other force |
| W_field = −qΔV | work done by the field on the charge | the same physics, charge-only system |
| v = √(2|q||ΔV| / m) | from rest, m in kg | find the speed after crossing ΔV |
| U_E = kqQ/r, V = kQ/r | k = 8.99 × 10⁹ N·m²/C² | closest-approach problems |
| Graph | Uniform field (from rest) | Near a fixed point charge |
|---|---|---|
| U_E against position | straight line | 1/r curve |
| K against position | straight line from zero | total energy minus the 1/r curve |
| K against time | curves upward | not needed |
Assumptions behind the numbers
- The source charges are fixed (or so massive they barely move).
- No energy leaves by radiation, friction or heating.
- Speeds are well below the speed of light, so K = ½mv².
- Gravity is ignored unless the object is large or the question says to include it.
Mistakes to avoid
- Dropping the sign of q. A negative charge has lower U_E at higher V.
- Confusing V with U_E. Volts are not joules.
- Double counting. Use W_field or ΔU_E, never both.
- Thinking distance matters. Same ΔV, same ΔK, whatever the gap.
- Assuming the heavier particle gains less energy. ΔK depends on q, not m.
Quick self-check
- A −5.0 nC charge moves from −20 V to +30 V. What is ΔU_E? (−2.5 × 10⁻⁷ J)
- A proton starts from rest and crosses 100 V. What are its K and speed? (1.6 × 10⁻¹⁷ J; 1.38 × 10⁵ m/s)
- An electron and a proton cross the same ΔV from rest. Which gains more kinetic energy? (Neither: same |q|, so same K. The electron is faster.)
Next: practice questions.