Question

In the given figure, sphere $1$ is a conducting sphere while sphere $2$ is a non-conducting solid sphere of radius $a$ and $b$ respectively. The distance between centres is $d$. Find the total energy of the arrangement. $(d>>a,d>>b)$

• A. $\frac{Q_1^2}{8\pi {\epsilon }_{0}a}+\frac{3Q_2^2}{20\pi {\epsilon }_{0}b}-\frac{Q_1{Q}_2}{4\pi {\epsilon }_{0}d}$
• B. $\frac{Q_2^2}{8\pi {\epsilon }_{0}a}+\frac{3Q_1^2}{8\pi {\epsilon }_{0}b}-\frac{Q_1{Q}_2}{8\pi {\epsilon }_{0}d}$
• C. $\frac{Q_2^2}{4\pi {\epsilon }_{0}a}+\frac{3Q_1^2}{8\pi {\epsilon }_{0}b}-\frac{Q_1{Q}_2}{8\pi {\epsilon }_{0}d}$
• D. None of above

Answer 1

The correct option is A $\frac{Q_1^2}{8\pi {\epsilon }_{0}a}+\frac{3Q_2^2}{20\pi {\epsilon }_{0}b}-\frac{Q_1{Q}_2}{4\pi {\epsilon }_{0}d}$

Self electrostatic potential energy of the sphere $1$ is,
$(U_{self}{)}_1=\frac{Q_1^2}{8\pi {\epsilon }_{0}a}$

Self electrostatic potential energy of the sphere $2$ is,
$(U_{self}{)}_2=\frac{3Q_2^2}{20\pi {\epsilon }_{0}b}$

Energy of the system due interaction between the charges distributed on sphere $1$ and $2$,

$\text{Interaction energy}=-\frac{Q_1{Q}_2}{4\pi {\epsilon }_{0}d}$

Now,
Total energy$U_t=(U_{self}{)}_1+(U_{self}{)}_2+\text{Interaction energy}$

Substituting the values,
$\therefore U_t=\frac{Q_1^2}{8\pi {\epsilon }_{0}a}+\frac{3Q_2^2}{20\pi {\epsilon }_{0}a}-\frac{Q_1{Q}_2}{4\pi {\epsilon }_{0}d}$

Hence, option (a) is correct answer.