Question

A non conducting sphere of radius a has a net charge $+q$ uniformly distributed throughout its volume. A spherical conducting shell having inner and outer radii $b$ and $c$ and net charge $- q$ is concentric with the sphere (see the figure).

Read the following statements
(i) The electric field at a distance $r$ from the center of the sphere for $r < a$ is $\frac{1}{4\pi\varepsilon_{0}} \frac{qr}{a^{3}}$
(ii) The electric field at distance $r$ for $a < r < b$ is $0$
(iii) The electric field at distance $r$ for $b < r < c$ is $0$
(iv) The charge on the inner surface of the spherical shell is $- q$
(v) The charge on the outer surface of the spherical shell is $+ q$
Which of the above statements are true?

• A. $(i)$, $(ii)$ and $(v)$
• B. $(i)$, $(iii)$ and $(iv)$
• C. $(ii)$, $(iii)$ and $(iv)$
• D. $(ii)$, $(iii)$ and $(v)$

Answer 1

Answer: (B)

Let $q ^{1}$ is the inner surface charge.
then, $q_{\text {outer }}=-q-q^{1}$ such that
$q_{\text {inner }}+q_{\text {outer }}=-q$
Electric field inside
the conductor should
be zero $(r > b$ and $r < e)$
so in the arbitrary
Gaussian surface
Electric flux $=\frac{\text { qinside }}{\varepsilon_{0}}=0$
$\Rightarrow q _{\text {inside }}( b < r < c )= q ^{1}+ q =0 $
$q ^{1}=- q$
So, $q _{\text {inner }}=- q$ and $q _{\text {outer }}=0$
(i) Electric field for $r<$ a ia,
$E .4 \pi r ^{2}=\frac{ q _{\text {inside }}}{\varepsilon_{0}}$
As a charge is uniformly distributed
$q$ inside $=\left(\frac{r^{3}}{a^{3}}\right) q$
So, $E .4 \pi r ^{2}=\frac{ r ^{3} q }{ a ^{3} \varepsilon_{0}}$
$\Rightarrow E =\frac{1}{4 \pi} \times \frac{ q }{ a ^{3} \varepsilon_{0}}$
(ii) Electric field for $a < r < b$,
$E=\frac{k q}{r^{2}}$
(iii) Electric field for $b < r < c$
As told earlier, $E = O$
(iv) Electric field for $c < r$
$E =\frac{ k _{\text {qouter }}}{ r ^{2}}= k \frac{(0)^{2}}{ r ^{2}}=0$