AP® Physics C: Electricity and Magnetism · Unit 9 · Topic 9.3
Energy Conservation for a Moving Charge
When a charge moves through a potential difference, its electric potential energy qV converts directly into kinetic energy. Energy conservation gives the speed without ever integrating force over distance.
A proton released near a positive plate feels a force the whole way across, so by Newton's second law it keeps accelerating as long as that force acts. You could find its final speed by integrating F = qE over the path, but there's a shortcut: energy conservation. The electric field does work on the charge, and that work equals the drop in potential energy, qV_initial - qV_final. Since no other forces act, that lost potential energy becomes kinetic energy. Set qDeltaV equal to (1/2)mv2 and solve for v. No force integral, no path details — just the start and end potentials.
This works because the electric force is conservative, exactly like gravity. Just as mgh converts cleanly into (1/2)mv2 for a falling ball, qV converts into (1/2)mv2 for a moving charge. The sign of q matters: a positive charge loses potential energy moving to lower V and speeds up, but a negative charge moving to lower V actually gains potential energy and slows down. The direction of motion relative to the field, not just the potential gradient, decides the outcome for negative charges.
This is the payoff of trading field vectors for potential, the whole theme of this unit. Instead of tracking force direction and magnitude step by step, you track one scalar number, V, at two points, and energy conservation does the rest. It connects directly to the next idea: equipotential surfaces, where a charge can move freely without any energy cost at all because V doesn't change.
Key terms
- Electric potential energy
- The energy U = qV a charge has because of its position in a potential field.
- Potential difference
- The change in potential, deltaV, a charge moves through, measured in volts.
- Conservative force
- A force, like the electric force, whose work depends only on start and end points, not the path taken.
- Distance of closest approach
- The point where an incoming charged particle's kinetic energy has fully converted to potential energy and it momentarily stops.
Watch it explained
- How it worksHow does a drop in qV turn straight into kinetic energy with no force integral needed?
- Worked exampleFind the speed of a proton starting from rest after falling through 500 volts.
- Worked exampleFind how close a 5 MeV alpha particle gets to a gold nucleus before it stops dead.
- Common mistakeWhy doesn't a charge always speed up when it moves toward lower potential?
Check yourself
End of topic.