Question

A hollow conducting sphere of radius R carries a negative charge -q. Give expressions for the magnitude of the electric field E inside (r < R) and outside (r > R) the sphere. (Use any variable or symbol stated above along with the following as necessary: k for the Coulomb constant.) Also indicate the direction of the field. Sketch a graph of the field strength as a function of r.

Answer 1

a)

To solve this question, use Gauss' Law, which states that the total electric flux (F) through a closed surface (called a "Gaussian surface") that contains within it a charge, q, is given by:

F = q/ε₀

where ε₀ is the permittivity.

The flux though a surface is given by the surface integral of the dot product of the electric field and the unit normal to the surface. In this case, the electric field is spherically symmetric, so if we use a spherical Gaussian surface, this integral reduces to

F = A * E

where E is the magnitude of the electric field, and A is the area of the surface.

Putting these two equations together gives:

E = q/(ε₀ * A)

which expresses the electric field as a function of the charge inside a closed surface and the area of that surface.

The surface area of a sphere with radius r (our Gaussian surface) is:

A(r) = 4*π*r²,

so we can write:

E(r) = q/(4ε₀*π*r²)

For E_inside, take the closed Gaussian surface to be a sphere with radius r < R (the radius of the charged shell). In this case, the charge enclosed by the Gaussian surface is zero because all the charge lies on the sperical shell, which is *outside our closed surface. The conclusion is that the electric field inside a charged, conducting shell is exactly zero everywhere inside the shell. More generally, the electric field inside any closed hollow conductor is zero, if the region enclosed by the conductor contains no charges.

so E_inside = 0

For E_outside, take the closed surface to be a sphere with radius r > R. The total charge contained inside this surface is simply the total charge on the charged sphere, -q, and:

E(r) = -q/(4ε₀*π*r²)