Question

Two spherical conductors A and B of radius a and b (b>a) are placed in air concentrically B is given charge +Q coulomb and A is grounded. The equivalent of these is:

• A. $4\pi {\epsilon }_0\frac{ab}{b-a}$
• B. $4\pi {\epsilon }_0\left(a+b\right)$
• C. $4\pi {\epsilon }_{0}b$
• D. $4\pi {\epsilon }_0\frac{b^2}{b-a}$

Answer 1

Answer: (A)

$4\pi {\epsilon }_0\frac{ab}{b-a}$

Electric field is non-vashing only in region $a<r<h$

Using Gauss's law

${\varphi }_s\overrightarrow{E}.\overrightarrow{dA}=E_{r}A=E_r\left(4\pi r^2\right)=\frac{Q}{{ϵ}_0}{E}_r=\frac{1}{4\pi {ϵ}_0}\left(\frac{Q}{r^2}\right)$

$\therefore$ Potential difference

$\Delta V=V_b-V_a=-{\int }_a^{b}Erdr=-\frac{Q}{4\pi {ϵ}_0}(\frac{1}{a}-\frac{1}{b})=\frac{Q}{4\pi {ϵ}_0}\left(\frac{b-a}{ab}\right)$

which yields

$C=\frac{Q}{|\Delta V|}=4\pi {ϵ}_0\left(\frac{ab}{b-a}\right)$