Charges on Conductors

Charge Inside a Cavity

When we put a charge , fixed in place, inside the cavity of our conductor, it induces a negative charge on the inside and a positive charge on the outer surface of the conductor. The inner surface will maintain and , while the outer surface will have net charge .

Figure

Grounded Conductor

What happens when we ground the conductor? Then we can supply the conductor with an infinite number of positive OR negative charges. It will supply these charges as needed. Furthermore, the potential of the Earth is .

Figure

Charges Outside a Conductor

An external charge on a conductor will induce an opposite charge on one side, and a like charge on the other side.

Figure

Grounded Conductor

Figure

Charges Released Into a Conductor

Suppose we touch a conductor with a charged rod, and release a charge or two into it. Then charge must distribute at the surface. More precisely, a non-uniform charge density will form (unless the shape is symmetrical), with "sharper" points having larger density and "flatter" areas having smaller density.

When a conductor with charge is connected with a neutral conductor via a thin, negligible, the charges will distribute amongst the conductors.

Figure

Without the connection, this diagram would look like #Charges Outside a Conductor. However, the charges are now free to move and distribute.

The left conductor had more protons than electrons. After connection, it gave away some of those protons to the bigger conductor, but still having a net positive number of protons.

Note

If we release a charge INSIDE a cavity, then that charge will also move the surface, while creating a net positive charge.

Boundary Conditions

Boundary conditions define what happen at the boundary of two different materials (mediums).

Think about like light getting refracted from air to water.

If the incident E-field is approaching at an angle to the new medium, there is a normal and tangential component to the vector.

Since the surface is equipotential, the tangential component will always be 0. For the normal component, we can use Gauss's law and find that .