Question 221: Electrostats & Capacitance

Question

Consider a uniform spherical charge distribution of radius $${R_1}$$ centred at the origin $$O.$$ In this distribution, a spherical cavity of radius $${R_2},$$ centred at $$P$$ with distance $$OP=a$$ $$ = {R_1} - {R_2}$$ (see figure) is made. If the electric field inside the cavity at position $$\overrightarrow r $$ is $$\overrightarrow E \overrightarrow {\left( r \right)} ,$$ then the correct statement(s) is (are)

Accepted Answer

Let $$\rho$$ be the charge density of the spherical charge distribution of radius r
1
centred at the origin O. A spherical cavity of radius r
2
centred at P with distance OP = a = r
1
$$-$$ r
2
is made in the spherical charge distribution.
The sphere with cavity is equivalent to a sphere of uniform charge density $$-$$$$\rho$$ and radius r
2
centred at P embedded in the original sphere. Thus, the electric field at a point Q in the cavity is superposition of (i) electric field at Q due to the sphere of charge density $$\rho$$ and radius r
1
centred at O (say $$\overrightarrow E $$
1
), and (ii) electric field at Q due to the sphere of charge density $$-$$$$\rho$$ and radius r
2
centred at P (say $$\overrightarrow E $$
2
). Let $$\overrightarrow a $$, $$\overrightarrow r $$, and $$\overrightarrow r $$ $$-$$ $$\overrightarrow a $$ be the vectors as shown in the figure. The electric fields $$\overrightarrow E $$
1
, $$\overrightarrow E $$
2
, and their superposition $$\overrightarrow E $$
12
are given by
$${\overrightarrow E _1} = {1 \over {4\pi { \in _0}}}{{{4 \over 3}\pi |\overrightarrow r {|^3}\rho } \over {|\overrightarrow r {|^2}}}\widehat r = {\rho  \over {3{ \in _0}}}\overrightarrow r $$,
$${\overrightarrow E _2} =  - {\rho  \over {3{ \in _0}}}(\overrightarrow r  - \overrightarrow a )$$,
$${\overrightarrow E _{12}} = {\overrightarrow E _1} + {\overrightarrow E _2} = {\rho  \over {3{ \in _0}}}\overrightarrow a $$.
Thus, the electric field at a point within the cavity is uniform and its magnitude and direction both depend on $$\overrightarrow a $$.

Answer

$$\overrightarrow E $$ is uniform, its magnitude is independent of $${R_2}$$ but its direction depends on $$\overrightarrow r .$$

Answer

$$\overrightarrow E $$ is uniform, its magnitude depends on $${R_2}$$ and its direction depends on $$\overrightarrow r .$$

Answer

$$\overrightarrow E $$ is uniform, its magnitude is independent of a but its direction depends on $$\overrightarrow a $$

Answer

$$\overrightarrow E $$ is uniform and both its magnitude and direction depend on $$\overrightarrow a $$

Electrostats & Capacitance

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