Chapter 21 · Free playable model
Potential Climber
Cross the field without running out of energy. You may only stand where K = E − qV is still positive.
ΔU = qΔV — drag the charge or take a step to see the transaction.
One positive source domes the middle of the field. Walk around the dome to the goal.
- Simple definition
- Electric potential is the electric potential energy a charge would have at a point, divided by that charge.
- Example
- Carrying +3 nC through a rise of 4 kV costs 12 µJ, whatever route you take between the two points.
How the physics works
V = ΣkQᵢ/rᵢ. Potential is a scalar, so each source contributes kQ/r with its own sign and the contributions simply add. With k = 8.99 × 10⁹ N·m²/C², a few hundred nanocoulombs at tens of centimetres gives a few kilovolts — the numbers printed beside each source.
U = qV and ΔU = qΔV. Standing at a point costs qV; moving between two points costs q times the potential difference and nothing else, so the route between them is free. The field does W = −qΔV on the carrier, and that work is exactly the change in kinetic energy. That is why the bill on a sketched route only ever quotes the endpoints.
K = E − qV > 0. Total energy is conserved, so kinetic energy depends only on where you stand. The region you may enter is the set of points where qV is smaller than E, bounded by the single equipotential V = E/q. That boundary is the hatched edge on the map.
Signs matter more than sizes. Flip the carrier from +3 nC to −3 nC and every price flips with it: positive sources become valleys and negative sources become walls. Level 03 exists to make that swap unmissable.
Capacitor energy, ½C(ΔV)², belongs to its own lesson rather than this game — see Parallel Plates and Capacitors.