A coronal discharge occurs when the electric field magnitude just above the surface of a conductor exceeds approximately \(3 \times 10^{6} \mathrm{~N} / \mathrm{C}\), a magnitude strong enough to ionize atoms in the air by ripping electrons from them. Coronal discharge is most likely to occur near a sharp tip of a conductor. (This is why solder joints in electronics must not have sharp protrusions and why it's easier to draw a spark from your fingertip than from your knee after scuffing your feet on a carpet.)

(a) Make an argument based on the geometry of the field lines emerging from a conducting surface that charge is more likely to concentrate on a convex region of the surface than on a concave region.

(b) Sketch a paring knife that has a surplus charge on it, and shade the regions where charge is most likely to concentrate on the surface.