Charge on a Conductor in Equilibrium

Dump extra charge on a metal block and the free electrons inside are mobile. If any electric field existed inside the metal, it would push those electrons, so charge keeps sliding around until the field inside the conductor is zero everywhere. That's the definition of equilibrium: charge stops moving. Once the interior field is zero, draw a Gaussian surface just inside the conductor's actual surface, entirely within the metal. Since the field is zero on that whole surface, the flux through it is zero, so by Gauss's law the enclosed charge must also be zero. That has to hold for any such surface you draw inside the metal, no matter how close to the outer edge. The only place left for the excess charge to live is the outer surface itself.

This kills the natural guess that charge spreads evenly through the bulk like sugar stirred into water. Charge isn't distributed through the volume at all in equilibrium — it's a surface phenomenon. The amount per unit area, the surface charge density sigma, can still vary across that surface even though the interior field is uniformly zero.

That variation sets up the next idea: sigma gets large where the surface curves sharply, like at a point or edge, because charge packs in tighter where there's less room to spread out. That's why lightning rods and sharp conductors concentrate field and charge, a preview of how shape governs capacitance and field behavior in the rest of the unit.