Question

The figure shows two conducting concentric sperical shells of radii R and 2R.
The inner shell has a charge +Q and the outer one is neutral.

(A)	Energy stored in region given by R < r < 2R	P) $\frac{Q^2}{32\pi {ϵ}_{o}R}$
(B)	Energy stored in region given by $2R<r<\infty$	Q) $\frac{Q^2}{16\pi {ϵ}_{o}R}$
(C)	Energy stored in region given by 2R < r < 4R	R) $\frac{Q^2}{8\pi {ϵ}_{o}R}$
(D)	Heat generated when switch S is closed	S) zero

T) None

• A. A $\to$ Q, B $\to$ Q, C $\to$ P, D $\to$ Q,
• B. A $\to$ R, B $\to$ Q, C $\to$ S, D $\to$ S,
• C. A $\to$ R, B $\to$ R, C $\to$ P, D $\to$ Q,
• D. A $\to$ Q, B $\to$ R, C $\to$ S, D $\to$ S,

Answer 1

The correct option is A A $\to$ Q, B $\to$ Q, C $\to$ P, D $\to$ Q,

Energy stored in region is
du = $\frac{1}{2}{ϵ}_o{E}^2\times 4\pi r^{2}dr$
where E = $\frac{Q}{4\pi {ϵ}_o{r}^2}$
$\cup =\int du=\int \frac{1}{2}{ϵ}_o(\frac{Q}{4\pi {ϵ}_o{r}^2}{)}^2\times 4\pi r^{2}dr$
$\cup =\frac{Q^2}{8\pi {ϵ}_o}{\int }_{r_1}^{r_2}{r}^{-2}dr=\frac{Q^2}{8\pi {ϵ}_o}[\frac{1}{r_1}-\frac{1}{r_2}]$

Varying $r_1$ & $r_2$ energies for different regions can be found.