Electrostatics 7 Question 19
21. 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
(2013 Adv.)
(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.
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Solution:
For electrostatic field,
$$ \begin{aligned} \mathbf{E} _P=\mathbf{E} _1+\mathbf{E} _2 & =\frac{\rho}{3 \varepsilon _0} \mathbf{C} _1 \mathbf{P}+\frac{(-\rho)}{3 \varepsilon _0} \mathbf{C} _1 \mathbf{P} \\ & =\frac{\rho}{3 \varepsilon _0}\left(\mathbf{C} _1 \mathbf{P}+\mathbf{P} \mathbf{C} _2\right) \\ \mathbf{E} _P & =\frac{\rho}{3 \varepsilon _0} \mathbf{C} _1 \mathbf{C} _2 \end{aligned} $$
