Question 629: Electrostats & Capacitance

Question

The region between two concentric spheres of radii $$'a'$$ and $$'b',$$ respectively (see figure), have volume charge density $$\rho  = {A \over r},$$ where $$A$$ is a constant and $$r$$ is the distance from the center. A such that the electric field in the region between the spheres will be constant, is :

Accepted Answer

Applying Gauss's law
$$\oint {s\overrightarrow E .\overrightarrow {ds} }  = {Q \over {{ \in _0}}}$$
$$	herefore$$ $$E 	imes 4\pi {r^2} = {{Q + 4\pi a{r^2} - 4\pi A{a^2}} \over {{ \in _0}}}$$
$$ho  = {{dr} \over {dv}}$$
$$Q = ho 4\pi {r^2}$$
$$Q = \int\limits_a {{A \over r}} 4{\pi ^2}dr = 4\pi A\,\left[ {{r^2} - {a^2}} ight]$$
$$E = {1 \over {4\pi { \in _0}}}\left[ {{{Q - 4\pi A{a^2}} \over {{r^2}}} + 4\pi A} ight]$$
For $$E$$ to be independent of $$'r'$$
$$Q - 2\pi A{a^2} = 0$$
$$	herefore$$ $$A = {Q \over {2\pi {a^2}}}$$

Answer

$${{2Q} \over {\pi \left( {{a^2} - {b^2}} \right)}}$$

Answer

$${{2Q} \over {\pi \,{a^2}}}$$

Answer

$${Q \over {2\pi \,{a^2}}}$$

Answer

$${Q \over {2\pi \,\left( {{b^2} - {a^2}} \right)}}$$

Electrostats & Capacitance

The region between two concentric spheres of radii $'a' and 'b', respectively (see figure), have volume charge density \rho = {A \over r}, where A is a constant and r$ is the distance from the center. A such that the electric field in the region between the spheres will be constant, is :

Choose the correct answer: