Question

Find the electric field E⃗ (r⃗ )E→(r→)E_vec(r_vec) inside the sphere (for rrr< aaa) in terms of the position vector r⃗ r→r_vec. Express your answer in terms of r⃗ r→r_vec, rhorhorho (Greek letter rho), and ϵ0ϵ0epsilon_0.

Answer 1

Solution :

Using the Gauss law, the electric field intensity of a sphere is given by :

$E. 4 \pi r^2 = \frac{Q_{enc}}{\epsilon_0}$

$E = \frac{Q_{enc}}{4 \pi r^2 \epsilon_0}$

Now the enclosed charge inside the sphere (r<a) is

$Q_{enc}= \rho V$

$Q_{enc}= \rho \left( \frac{4}{3} \pi r^3 \right)$

Hence, the electric field intensity becomes as follows :

$E= \frac{\rho \left(\frac{4}{3} \pi r^3\right)}{4 \pi \epsilon_0 r^2}$

$E =\frac{\rho r}{3 \epsilon_0}$

Thus, the electric field inside the sphere is given by :

$\vec {E} = \frac{\rho \vec{r}}{3 \epsilon_0}$