Question 233: Electrostats & Capacitance

Question

A spherical metal shell A of radius $${R_A}$$ and a solid metal sphere $$B$$ of radius $${R_B}\left( { < {R_A}} \right)$$ are kept far apart and each is given charge $$' + Q'.$$ Now they are connected by a thin metal wire. Then

Accepted Answer

When connected by a wire, charges on A and B are redistributed until potential on both becomes equal. After the charge redistribution, A and B are not influenced by each other because they are far apart. Thus, $$E_A^{inside} = 0$$ as field inside a conducting shell is zero.
So choice (a) is correct.
Let Q
A
and Q
B
are the charges on metal shell A and metal sphere B after they are connected by a wire. Since their electric potentials will be equal,
V
A
= V
B
$$ \Rightarrow {{{Q_A}} \over {4\pi {\varepsilon _0}{R_A}}} = {{{Q_B}} \over {4\pi {\varepsilon _0}{R_B}}} \Rightarrow {{{Q_A}} \over {{Q_B}}} = {{{R_A}} \over {{R_B}}}$$
Since $${R_B}  {Q_B}$$. So choice (b) is correct.
Now, $${\sigma _A} = {{{Q_A}} \over {4\pi R_A^2}}$$ and $${\sigma _B} = {{{Q_B}} \over {4\pi R_B^2}}$$
$$	herefore$$ $${{{\sigma _A}} \over {{\sigma _B}}} = {{{Q_A}} \over {{Q_B}}} 	imes {\left( {{{{R_B}} \over {{R_A}}}} ight)^2} = {{{R_A}} \over {{R_B}}} 	imes {\left( {{{{R_B}} \over {{R_A}}}} ight)^2} = {{{R_B}} \over {{R_A}}}$$
Hence, choice (c) is also correct.
Electric fields on the surface of shell and sphere are
$${E_A} = {{{\sigma _A}} \over {{\varepsilon _0}}}$$
and $${E_B} = {{{\sigma _B}} \over {{\varepsilon _0}}}$$
$$	herefore$$ $${{{E_A}} \over {{E_B}}} = {{{\sigma _A}} \over {{\sigma _B}}} < 1$$, i.e. $${E_A} < {E_B}$$
So, choice (d) is also correct. All the four choices are correct.

Answer

$$E_A^{inside} = 0$$

Answer

$${Q_A} > {Q_B}$$

Answer

$${{{\sigma _A}} \over {{\sigma _B}}} = {{{R_B}} \over {{R_A}}}$$

Answer

$$E_A^{on\,\,surface} < \,E_B^{on\,\,surface}$$

Electrostats & Capacitance

A spherical metal shell A of radius ${R_A} and a solid metal sphere B of radius {R_B}\left( { < {R_A}} \right) are kept far apart and each is given charge ' + Q'.$ Now they are connected by a thin metal wire. Then

Choose the correct answer: