Question

Imagine that you have a conducting sphere. Draw a picture that shows the charge distribution within the shell and briefly describe a reason or reasons for the electric field inside of the shell is zero using Gauss law. Please explain thoroughly and write neatly!

Answer 1

When a charge is given to a conductor ,it produces it’ s own field in side the conductor. The free electron charges of the conductor are influenced by this field and they start moving. Thus a current is established in the conductor. But, for static equilibrium of charges the current inside the conductor must be zero and hence, extra charge given to the conductor can not remain inside a conductor and it has to move on the outer surface of the conductor. Thus, any charge given to the hollow sphere of conductor remains on the outer surface of the sphere. Now, assuming that there is no charge inside hollow space enclosed by hollow sphere and knowing that outer surface is equipotential or field is in the local perpendicular direction or there is uniform surface distribution of charges we can take the field out side the hollow sphere as having spherical symmetry. This suggests that for out side field we take spherical surface as Gaussian surface.

Let us take a spherical Gaussian surface of radius R(where,R>r). r is radius of hollow sphere. If E is field at R, then according to Gauss’s theorem surface integral E.da over Gaussian surface is equal to charge Q enclosed by the Gaussian surface divided by epsilon zero.

Here,da is surface element vector.

But, E is constant on our chosen Gaussian surface. Therefore E can be taken out from surface integral and then integral da over the surface is

4 pi R^2. Thus,

E4 pi R^2=Q/(epsilon zero)

Or

E=Q/4 pi epsilon zero R^2. This is the field out side the given hollow sphere. This field is as if the whole charge on the outer surface is concentrated on the center of the hollow sphere.

To find field inside the hollow sphere we can take take Gaussian surfaces with radii ranging from R=0 to R=r. Any of these surfaces the charge enclosed is zero and hence field at all the points on these surfaces is zero. Thus, electric field in side a hollow sphere of conductor is zero.