Question

Two non-conducting spheres of radii $R_1$ and $R_2$ and 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 different direction

Answer 1

Answer: (C)

the electrostatic field is constant in magnitude

In triangle $P{C}_1{C}_2$

$\overrightarrow{r_2}=\overrightarrow{d}=\overrightarrow{r_1}$

The electrostatic field at point $P$ is

$\overrightarrow{E}=\frac{K\left(\rho \frac{4}{3}\pi {R_1}^3\right)\overrightarrow{r_2}}{{R_1}^3}+\frac{K\left(\rho \frac{4}{3}\pi {R_2}^3\right)\left(\overrightarrow{-r_1}\right)}{{R_2}^3}$

$\overrightarrow{E}=K\rho \frac{4}{3}\pi (\overrightarrow{r_2}-\overrightarrow{r_1})$

$\overrightarrow{E}=\frac{\rho }{3{ϵ}_0}\overrightarrow{d}$

Hence, option (C) is correct.