We know from the example in Section 111 that the electric fi

We know from the example in Section 1.11 that the electric field inside a solid sphere with uniform charge density is proportional to r. Assume instead that the charge density is not uniform, but depends only on r. What should this dependence be so that the magnitude of the field at points inside the sphere is independent of r(except right at the center, where it isn\'t well defined)? What should the dependence be in the analogous case where we have a cylinder instead of a sphere?

Solution

Electric field inside a sphere of radius R, charge Q at a point at distance r from the centre of sphere inside it:
E = kQr/R^3 [ k = coulomb\'s constant]

for this to be independent of r, let Q = Qo/r and hence E = kQo/R^3 = independent of r

Electric field inside a cylinder of radius R, hieght l, charge Q at a point inside the cylinder at distace r from the axis of the cylinder:
E = 2kQr/l*R^2

for this to be independent of r, let Q = Qo/r and E = 2kQo/l*r^2

i.e. the same dependency