Define electric potential at a point. Derive an expression for the potential at a distance r from a point charge q.

1. 1 · Define. The potential at a point is the work done against the field in bringing a unit positive charge from infinity to that point. Unit: volt.
2. 2 · Draw the figure. [Charge +q at O; a straight line going out through P, A, B and C towards infinity, with OP = r]
3. 3 · Work in one small step. From A to P, take the average field as kq/(r_A r). Work = kqq₀(1/r − 1/r_A).
4. 4 · Add all the steps. The middle terms cancel: W = kqq₀(1/r − 1/∞) = kqq₀/r.
5. 5 · Divide by q₀. V = W/q₀ = (1/4πε₀) q/r.
6. 6 · Close in one line. So the potential of a point charge is a scalar that falls as 1/r and has the sign of q.

Scoring tip: Write 'unit positive charge' and 'from infinity' in the definition. Both phrases are marked.