Question

A conducting shell $S_1$ having a charge $Q$ is surrounded by a uncharged concentric conducing spherical shall $S_2$. Let the potential difference between $S_1$ and that $S_2$ be $V.$ If the shall $S_2$ is now given a charge $-3Q,$ then new potential different between the same two shall is:

• A. $V$
• B. $2V$
• C. $4V$
• D. $-2V$

Answer 1

The correct option is C $4V$

For case $I$ the potential between the two shells is

${\int }_{r_1}^{r_2}\frac{Q}{4\pi {ϵ}_0{r}^2}dx$

$=\frac{Q}{4\pi {ϵ}_0}[\frac{1}{x_1}-\frac{1}{x_2}]$

According to the questions for the $1^{st}$ case, this is $V$

$\therefore \frac{Q}{4\pi {ϵ}_0}[\frac{1}{r_1}-\frac{1}{r_2}]=V$

For the $2^{nd}$ case from pietere $II$, potential difference

${\int }_{r_1}^{r_2}\frac{Q}{4\pi {ϵ}_0{r}^2}dx+{\int }_{r_2}^{r_1}\frac{(-3Q)}{4\pi {ϵ}_0{r}^2}dx$

$=4{\int }_{r_1}^{r_2}\frac{Q}{4\pi {ϵ}_0{r}^2}dx$

$=4\frac{Q}{4\pi {ϵ}_0}[\frac{1}{r_1}-\frac{1}{r_2}]$

$4V$