Question 178 of 311

Three concentric spherical metallic shells $X,Y$ and $Z$ of radius $a,b$ and $c$ respectively $\left[a\lt b\lt c\right]$ have surface charge densities $\sigma ,–\sigma$ and $\sigma$, respectively. The shells $X$ and $Z$ are at same potential. If the radii of X&Y are $2cm$ and $3cm$, respectively. The radius of shell $Z$ is _____$cm$.

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Answer: 5

The charges on the spheres are
${q}_{X}=\sigma 4\pi {a}^{2} {q}_{Y}=-\sigma 4\pi {b}^{2} {q}_{Z}=\sigma 4\pi {c}^{2}$
It is given that ${V}_{X}={V}_{Z}$
So,
$\dfrac{{q}_{X}}{4\pi {\epsilon }_{0}a}+\dfrac{{q}_{Y}}{4\pi {\epsilon }_{0}b}+\dfrac{{q}_{Z}}{4\pi {\epsilon }_{0}c}=\dfrac{{q}_{X}}{4\pi {\epsilon }_{0}c}+\dfrac{{q}_{Y}}{4\pi {\epsilon }_{0}c}+\dfrac{{q}_{Z}}{4\pi {\epsilon }_{0}c} \Rightarrow \dfrac{\sigma 4\pi {a}^{2}}{a}-\dfrac{\sigma 4\pi {b}^{2}}{b}+\dfrac{\sigma 4\pi {c}^{2}}{c}=\dfrac{\sigma 4\pi ({a}^{2}-{b}^{2}+{c}^{2})}{c} \Rightarrow c(a-b+c)=({a}^{2}-{b}^{2}+{c}^{2}) \Rightarrow c=a+b=2cm+3cm=5cm$