Question

Two metal spheres of radii $a$ and $b$ are connected by a thin wire. Their separation is very large compared to their dimensions. The capacitance of this system is

• A. $4\pi {ϵ}_0\left(ab\right)$
• B. $2\pi {ϵ}_0(a+b)$
• C. $4\pi {ϵ}_0(a+b)$
• D. None of these

Answer 1

The correct option is B $2\pi {ϵ}_0(a+b)$

For spherical capacitors capacitance is given by $\frac{4\pi {\epsilon }_0{R}_1{R}_2}{R_2-R_1}$

$=\frac{4\pi {\epsilon }_0{R}_1{R}_2}{R_2-R_1}$

Potential at $A=\frac{q_1}{4\pi {\epsilon }_0{r}_1}=\frac{q_1}{4\pi {\epsilon }_{0}a}=V$

Potential at $B=\frac{q_2}{4\pi {\epsilon }_0{r}_2}=\frac{q_2}{4\pi {\epsilon }_{0}b}=V$

Since $P.E.$ at $A=$ Potential at B.

Capacitance $=\frac{q_1+q_2}{2V}=\frac{q}{\frac{2\times q_1}{4\pi {\epsilon }_{0}q}}$

$q_1={\left[\frac{a}{a+b}\right]}^q$

$C=2\pi {\epsilon }_0(a+b)$