Electric Potential AP Physics Montwood High School R. Casao

Electric Potential of a Sphere Consider a conducting spherical shell of radius R. If the shell carries a charge +Q, the charge will lie on the outer surface of the shell and the electric field inside the sphere will be 0 N/C (Q in = 0 C).

Electric Potential of a Sphere At points outside the shell, the electric field exists as if all of the charge were concentrated at the center (like a point charge) for r > R

Electric Potential of a Sphere For points inside the shell: –Since the electric field = 0 N/C inside the shell, there is no electric force and no work done on a charge moving inside the shell. The potential is constant within the sphere. So if we move a charge q from the surface of the sphere to its inside, the potential in J/C would not change. The potential everywhere inside the shell is equal to the potential on the surface. for r < R

Electric Potential of a Sphere These two conditions also apply to a solid conducting sphere.

Electric Potential of a Nonconducting (Insulating) Sphere If a nonconducting (or insulating) sphere of radius R has a charge of +Q distributed uniformly throughout its volume (charge density  is constant). At points outside the sphere: –The electric field exists as if all of the charge were concentrated at the center (like a point charge) for r > R

Electric Potential of a Nonconducting (Insulating) Sphere At points inside the sphere, uniform charge density allows us to set up the proportional relationship b/w charge and volume to determine the charge contained within the sphere of radius r:

Electric Potential of a Nonconducting (Insulating) Sphere From Gauss’ Law :

Electric Potential of a Nonconducting (Insulating) Sphere Using : And moving from the outer surface R of the sphere inside to the location of radius r:

Electric Potential of a Nonconducting (Insulating) Sphere

Replace the V R equation: Add to both sides:

Electric Potential of a Nonconducting (Insulating) Sphere Multiply by 2·R 2 to get a common denominator of 2·R 3.

Electric Potential of a Nonconducting (Insulating) Sphere Factor out

The Potential of a Cylinder Consider a cylinder of radius R which carries a uniform charge density . The electric field at a distance r from the center of the cylinder is: In a problem like this, the potential is 0 V at a position some distance away from the cylinder. At r = a, V a = 0 V.

The Potential of a Cylinder Using Integrate from a to r.

The Potential of a Cylinder

Use the properties of logarithms (divide):