State Gauss's theorem. Use it to find the electric field due to a uniformly charged thin spherical shell at a point (a) outside and (b) inside the shell. Draw the graph of E against r.

1. 1 · State the theorem. The flux through any closed surface = q(enclosed)/ε₀.
2. 2 · Draw the shell. [Shell of radius R with charge q; a dashed Gaussian sphere of radius r > R outside and one of radius r < R inside; E arrows radially outward]
3. 3 · Use symmetry. E is radial and has the same size at every point of a concentric sphere, so the flux = E × 4πr².
4. 4 · Outside (r > R). Enclosed charge = q. E × 4πr² = q/ε₀, so E = q/(4πε₀r²), as if q were at the centre.
5. 5 · Inside (r < R). Enclosed charge = 0. E × 4πr² = 0, so E = 0.
6. 6 · Draw the graph. [E = 0 from 0 to R; a jump to q/(4πε₀R²) at R; then a 1/r² fall]
7. 7 · Real life. A closed metal body keeps its inside free of field, so a car or bus is a safe shelter during lightning.
8. 8 · Close in one line. A charged shell acts like a point charge outside it and gives no field inside it.

Scoring tip: Name the Gaussian surface and say why E is the same all over it (symmetry). That one sentence is often worth a full mark.