Question 159: Electrostatics & Capacitance

Question

There are three co-centric conducting spherical shells $A, B$ and $C$ of radii $a, b$ and $c$ respectively ( $c>b>a$ ) and they are charged with charge $q_1, q_2$ and $q_3$ respectively. The potentials of the spheres $A, B$ and $C$ respectively, are :

Accepted Answer

The electric potential V due to a charged spherical shell of radius R and charge Q is given by :
$$ \mathrm{V}=\left\{\begin{array}{l} \frac{1}{4 \pi \epsilon_0} \frac{\mathrm{Q}}{\mathrm{R}}, \mathrm{r}\mathrm{R} \end{array}\right. $$
Point A is on the surface of shell A, inside shell B, and inside shell C. The total potential is the sum of potentials due to each shell (Superposition Principle).
So, the potential at A ;
1. Due to A (Surface, $\mathrm{r}=\mathrm{a}) \Rightarrow \mathrm{V}_{\mathrm{AA}}=\frac{\mathrm{k} \mathrm{q}_1}{\mathrm{a}}$
2. Due to B (Inside, $\mathrm{r}<\mathrm{b}) \Rightarrow \mathrm{V}_{\mathrm{AB}}=\frac{\mathrm{kq}_2}{\mathrm{~b}}$
3. Due to C (Inside, $\mathrm{r}<\mathrm{c}) \Rightarrow \mathrm{V}_{\mathrm{AC}}=\frac{\mathrm{kq}_3}{\mathrm{c}}$
Hence, total potential at point A is,
$$ V_A=\left(\frac{\mathrm{kq}_1}{\mathrm{a}}+\frac{\mathrm{kq}_2}{\mathrm{~b}}+\frac{\mathrm{kq}_3}{\mathrm{c}}\right)=\frac{1}{4 \pi \epsilon_0}\left(\frac{\mathrm{q}_1}{\mathrm{a}}+\frac{\mathrm{q}_2}{\mathrm{~b}}+\frac{\mathrm{q}_3}{\mathrm{c}}\right) $$
For the point B which is outside shell A , on the surface of shell B , and inside shell C . So, the potential at point B,
1. Due to A (Outside, distance b$) \Rightarrow \mathrm{V}_{\mathrm{BA}}=\frac{\mathrm{k} \mathrm{q}_1}{\mathrm{~b}}$
2. Due to B (Surface, $\mathrm{r}=\mathrm{b}) \Rightarrow \mathrm{V}_{\mathrm{BB}}=\frac{\mathrm{kq}_2}{\mathrm{~b}}$
3. Due to C (Inside, $\mathrm{r}<\mathrm{c}) \Rightarrow \mathrm{V}_{\mathrm{BC}}=\frac{\mathrm{kq}_3}{\mathrm{c}}$
Hence, total potential at point B is:
$$ V_B=\frac{1}{4 \pi \epsilon_0}\left(\frac{q_1}{b}+\frac{q_2}{b}+\frac{q_3}{c}\right) $$
$$\Rightarrow $$ $$ V_B=\frac{1}{4 \pi \epsilon_0}\left(\frac{q_1+q_2}{b}+\frac{q_3}{c}\right) $$
Point C is outside shell A , outside shell B , and on the surface of shell C . So, potential at point C is,
1. Due to A (Outside, distance c$) \Rightarrow \mathrm{V}_{\mathrm{CA}}=\frac{\mathrm{kq}_1}{\mathrm{c}}$
2. Due to B (Outside, distance c$) \Rightarrow \mathrm{V}_{\mathrm{CB}}=\frac{\mathrm{kq}_2}{\mathrm{c}}$
3. Due to C (Surface, $\mathrm{r}=\mathrm{c}) \Rightarrow \mathrm{V}_{\mathrm{CC}}=\frac{\mathrm{kq}_3}{\mathrm{c}}$
Hence, total potential at point C is :
$$ V_C=\frac{1}{4 \pi \epsilon_0}\left(\frac{q_1}{c}+\frac{q_2}{c}+\frac{q_3}{c}\right) $$
$$\Rightarrow $$ $V_C=\frac{1}{4 \pi \epsilon_0}\left(\frac{q_1+q_2+q_3}{c}\right)$
So, $\mathrm{V}_{\mathrm{A}}=\frac{1}{4 \pi \epsilon_0}\left(\frac{\mathrm{q}_1}{\mathrm{a}}+\frac{\mathrm{q}_2}{\mathrm{~b}}+\frac{\mathrm{q}_3}{\mathrm{c}}\right), \mathrm{V}_{\mathrm{B}}=\frac{1}{4 \pi \epsilon_0}\left(\frac{\mathrm{q}_1+\mathrm{q}_2}{\mathrm{~b}}+\frac{\mathrm{q}_3}{\mathrm{c}}\right), \mathrm{V}_{\mathrm{C}}=\frac{1}{4 \pi \epsilon_0}\left(\frac{\mathrm{q}_1+\mathrm{q}_2+\mathrm{q}_3}{\mathrm{c}}\right)$.
Hence, the correct option is (D).

Answer

$$ \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1}{a}+\frac{q_2}{b}+\frac{q_3}{c}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2+q_3}{b}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2+q_3}{c}\right) $$

Answer

$$ \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2+q_3}{a}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2+q_3}{b}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2+q_3}{c}\right) $$

Answer

$$ \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2+q_3}{a}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2}{b}+\frac{q_3}{c}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1}{a}+\frac{q_2}{b}+\frac{q_3}{c}\right) $$

Answer

$$ \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1}{a}+\frac{q_2}{b}+\frac{q_3}{c}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2}{b}+\frac{q_3}{c}\right), \frac{1}{4 \pi \epsilon_{\mathrm{o}}}\left(\frac{q_1+q_2+q_3}{c}\right) $$

Electrostatics & Capacitance

There are three co-centric conducting spherical shells A, B and C of radii a, b and c respectively ( c>b>a ) and they are charged with charge q_1, q_2 and q_3 respectively. The potentials of the spheres A, B and C respectively, are :

Choose the correct answer: