Q.Explain in detail how charges are distributed in a conductor, and the principle behind the lightning conductor.

Step 1. Two connected spheres. Two conducting spheres of radii $r_1$, $r_2$ joined by a wire reach a common potential, $kq_1/r_1=kq_2/r_2$, giving $q_1/q_2=r_1/r_2$.

Step 2. Surface charge density. Since $\sigma=q/(4\pi r^2)$, this gives $\sigma_1/\sigma_2=r_2/r_1$ -- the smaller sphere carries the higher surface charge density.

Step 3. Generalising to an irregular conductor. Any single conductor of irregular shape can be thought of as locally spherical at every point, with its own local radius of curvature; the more sharply curved regions (points, edges) always develop the higher charge density, and hence the strongest local field, $E=\sigma/\varepsilon_0$.

Step 4. Action at points. If this local field exceeds air's dielectric strength, the surrounding air ionises and charge leaks away continuously -- corona discharge, or action at points. …

Derive the two-sphere charge-sharing result, generalise it to an irregular conductor's local curvature, then connect it to …

• Stopping at the two-sphere result without generalising to the action-at-points and lightning-conductor …