Question

A charge Q is distributed uniformly within the material of a hollow sphere of inner and outer radii $r_1$ and $r_2$ (figure 30-E4). Find the electric field at a point P a distance x away from the centre for $r_1<x<r_2$. Draw a rough showing the electric field as a function of x for $0<x<2r_2$ (figure 30-E4).

Answer 1

Let the arbitrary radius x.

The charge on sphere

$=\frac{\frac{4}{3}\pi (x^3-r_1^3)q}{\frac{4}{3}\pi (r_2^2-r_1^2)}$

$=\frac{(x^3-r_1^3)}{(r_2^3-r_1^3)q}$

According to Gauss's law

$\int E.ds=\frac{q}{{\in }_0}$

$E.4\pi x^2=\frac{(x^3-r_1^3)q}{(r_2^3-r_1^3){\in }_0}$

or $E=\frac{q(x^3-r_1^3)}{4\pi {\in }_0{x}^2(r_2^3-r_1^3)}$