Question

There are two non-conducting spheres having uniform volume charge densities $\rho$ and $-\rho$. Both spheres have equal radius $R$. The spheres are now laid down such that they overlap as shown in the figure.
The electric field $E$ in the overlap region is

• A. Uniform
• B. zero
• C. $\frac{\rho d}{3{ϵ}_0}$
• D. $\frac{\rho R}{3{ϵ}_0}$

Answer 1

Answer: (A), (C)

$\frac{\rho d}{3{ϵ}_0}$


Now let us consider a point $P$ in the overlapped region at a distance $\overrightarrow{r_1}$ from the centre of the first sphere and $\overrightarrow{r_2}$ from the centre of second sphere. Now the electric field at $P$ due to sphere charge density $-\rho$ will be,
$E_1=\frac{-\rho \overrightarrow{r_1}}{3{ϵ}_0}$
Now the electric field at $P$ due to sphere charge density $\rho$ will be,
$E_2=\frac{\rho \overrightarrow{r_2}}{3{ϵ}_0}$
Net field at P will be,
$E=E_1+E_2$
$E=\frac{-\rho \overrightarrow{r_1}}{3{ϵ}_0}+\frac{\rho \overrightarrow{r_2}}{3{ϵ}_0}$
But from the figure,
$\overrightarrow{d}=\overrightarrow{r_2}-\overrightarrow{r_1}$
Therefore, the electric field in the overlapping region will be,
$E=\frac{\rho \overrightarrow{d}}{3{ϵ}_0}$