Question

A charged conducting sphere $A$ having radius $a$ is brought into contact with an uncharged conducting sphere $B$ having radius $b.$ After the spheres are disconnected and separated by large distance, the energy stored in sphere $B$ is $U$. The charge held by sphere $A$ before making contact is

• A. $q=b\sqrt{\frac{8\pi {\epsilon }_{0}U}{b}}$
• B. $q=a\sqrt{\frac{8\pi {\epsilon }_{0}U}{b}}$
• C. $q=(a+b)\sqrt{\frac{8\pi {\epsilon }_{0}U}{b}}$
• D. $q=b\sqrt{\frac{4\pi {\epsilon }_{0}U}{b}}$

Answer 1

The correct option is A $q=b\sqrt{\frac{8\pi {\epsilon }_{0}U}{b}}$

Radius $=a$

Sphere $B$ having radius $=b$

$E=U$

$q_1.q_{2}A=D$

$F=\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{a^2}$

Putting value,

$F=\frac{1}{4\pi {ϵ}_0}\frac{Q/2Q/2}{r^2}$

$F=\frac{1}{4\pi {ϵ}_0}\frac{Q^2}{{4D}^2}$

$F=\frac{1}{4}\left(\frac{1}{4\pi {ϵ}_0}\frac{Q^2}{D^2}\right)$

or, $U=\frac{1}{16\pi {ϵ}_0}\frac{Q^2}{D^2}$ $\because$ $(D=b)$

or, $Q=\sqrt{\frac{8\pi {ϵ}_{0}U}{b}}$ for unchanged body.