Question

Two non-conducting spheres of radii $R_1$ and $R_2$ carrying uniform volume charge densities $+\rho$ and$-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region:

• A. the electrostatic field is zero
• B. the electrostatic potential is constant
• C. the electrostatic field is constant in magnitude
• D. the electrostatic field has same direction

Answer 1

Answer: (C), (D)

The correct options are
C the electrostatic field is constant in magnitude
D the electrostatic field has same direction

Let the vector joining the centers of the two spheres be $\overrightarrow{d}$

and consider a point $P$ in the overlap region, whose position vector be $\overrightarrow{r_1}$ and $\overrightarrow{r_2}$ with respect to the centers of each sphere respectively.

Thus, by triangle law of vector addition, $\overrightarrow{r_1}-\overrightarrow{r_2}=\overrightarrow{d}$

Electric field at $P$ due to the positively charged sphere is $E_1=\frac{1}{4\pi {ϵ}_0}\frac{q}{r_1^3}\overrightarrow{r_1}$

Here, $q$ is the total charge enclosed in the sphere of radius $r_1$

Thus, $q=\rho \times \frac{4}{3}\pi r_1^3$

Hence, $E_1=\frac{\rho }{3{ϵ}_0}\overrightarrow{r_1}$

Similarly, Electric field due to negatively charged sphere is $E_2=-\frac{\rho }{3{ϵ}_0}\overrightarrow{r_2}$

Thus, net electric field at $P$ is $E=E_1+E_2=\frac{\rho }{3{ϵ}_0}(\overrightarrow{r_1}-\overrightarrow{r_2})=\frac{\rho }{3{ϵ}_0}\overrightarrow{d}$

Both, direction and magnitude of the electric field are constant in the overlap region.