Question

Consider an assembly for three conducting concentric spherical shells of radii a, b and c as shown in figure. Suppose the space between the two inner shells is filled with a dielectric of the dielectric constant K. Find the capacitance of the system between A and B.

• A. $\frac{4\pi {ϵ}_0{b}^2}{(c-b)+K(b-a)}$
• B. $\frac{4\pi {ϵ}_{0}Kac}{(c-a)}$
• C. $\frac{4\pi {ϵ}_{0}Kbc}{(c-b)}$
• D. $\frac{4\pi {ϵ}_{0}Kabc}{Ka(c-b)+c(b-a)}$

Answer 1

The correct option is D $\frac{4\pi {ϵ}_{0}Kabc}{Ka(c-b)+c(b-a)}$

Capacitance of spherical shell capacitor, $C=4\pi ϵ\frac{R_{in}\times R_{out}}{R_{our}-R_{in}}$ , where $ϵ=$ permittivity of space in between spherical shell.

Dielectric constant , $K=\frac{ϵ}{{ϵ}_0}$, for air media $K=1$.

Capacitance in between shells of radii a and b, $C_1=4\pi K{ϵ}_0\frac{ab}{b-a}$

Capacitance in between shells of radii b and c, $C_2=4\pi {ϵ}_0\frac{bc}{c-b}$

Capacitance between A and B, $C_{eq}=\frac{C_1{C}_2}{C_1+C_2}$ (as $C_1,C_2$ are in series)

$C_{eq}=4\pi {ϵ}_{0}K\frac{\frac{a{b}^{2}c}{(b-a)(c-b)}}{\frac{Kab}{b-a}+\frac{bc}{c-b}}$

$=4\pi {ϵ}_{0}K\frac{a{b}^{2}c}{Kab(c-b)+bc(b-a)}=\frac{4\pi {ϵ}_{0}Kabc}{Ka(c-b)+c(b-a)}$