Question

A spherical conductor carries a charge −30 μC and has a radius of 20 cm.

a) Determine the electric field and potential at all points in space.

b) Calculate the values of the field and potential at the following points:

i) r = 20 cm, ii) r = 15 cm, iii) r = 40 cm.

Answer 1

(a) Let R is the radius of the sphere, and Q be the charge on the conductive sphere.

We can calculate the electric field from Gauss's Law.

And the Potential can be calculated using the following relation with electric field.

Case.1: For r>R

Electric Field:

Electric Potential:

Case.2: For r=R

Electric Field:

Electric Potential:

Case.3: For r<R

For a conductive sphere, the charge resides only on the surface, so, charge inside is zero.

so for r < R i.e., within the sphere, Q=0.

Electric Field,

  

And hence, the potential should be constant, as the gradient of the potential should be zero which is the electric field.

We can get the value as follows,

(b)

(i) r=20 cm; i.e, r=R

So, let us substitute the values of in E and V expressions as obtained above case.2

and potential,

(ii)r=15 cm; i.e, r<R

So, let us substitute the values of in E and V expressions as obtained above case.3

E=0

and the Potential is same as that of the value that is on surface.

(iii) r= 40 cm; i.e, r>R

So, let us substitute the values of in E and V expressions as obtained above case.1

and potential,