Conductors and Capacitors

In electrostatic equilibrium, extra charge on a conductor spreads over its surface, the field inside is zero, the field just outside is perpendicular to the surface, and every point is at the same potential. Charge crowds more densely at sharp points and edges, and a closed conducting shell shields the space inside it from outside fields.

When conductors touch or are joined by a wire, charge flows until they’re all at the same potential, so the total charge is shared but not always equally (a bigger sphere ends up with more). Ground is a huge reservoir at zero potential, so grounding a conductor while a charged object is nearby, then breaking the connection, leaves it with an induced charge opposite to the nearby charge.

A capacitor is two conductors holding equal and opposite charges, and its capacitance $C = \frac{Q}{\Delta V}$ depends only on its geometry and the material between them. For parallel plates $C = \frac{\kappa\varepsilon_0 A}{d}$ and the field between them is nearly uniform, so a charge in the gap moves like a projectile; the stored energy is $U_C = \frac{1}{2}Q\Delta V = \frac{1}{2}C(\Delta V)^2$. You also use Gauss’s law to find C for spherical and cylindrical capacitors.

A dielectric is an insulator that becomes polarized in an electric field, making its own field that partly cancels the outside one. Filling a capacitor with a dielectric of constant κ multiplies its capacitance by κ: if the capacitor is isolated (charge fixed), the field and voltage drop by a factor of κ, but if it stays connected to a battery (voltage fixed), more charge flows onto the plates.