Three concentric spherical metallic shells $X , Y$ and $Z$ of radius $a , b$ and c respectively $[ a < b < c ]$ have surface charge densities $\sigma,-\sigma$ and $\sigma$, respectively. The shells $X$ and $Z$ are at same potential. If the radii of $X$ and $Y$ are $2\,cm$ and $3\,cm$, respectively.The radius of shell $Z$ is $......cm$.
JEE MAIN 2023, Diffcult
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$q _{ x }=\sigma 4 \pi a ^2$
$q _{ y }=-\sigma 4 \pi b ^2$
$q _{ z }=\sigma 4 \pi c ^2$
Potential $x =$ potential $z$
$V _{ x }= V _{ z }$
$\frac{ q _{ x }}{4 \pi \varepsilon_0 a }+\frac{ q _{ y }}{4 \pi \varepsilon_0 b }+\frac{ q _{ z }}{4 \pi \varepsilon_0 c }=\frac{ q _{ x }}{4 \pi \varepsilon_0 c }+\frac{ q _{ y }}{4 \pi \varepsilon_0 c }+\frac{ q _{ z }}{4 \pi \varepsilon_0 c }$
$\frac{\sigma 4 \pi a ^2}{ a }-\frac{\sigma 4 \pi b ^2}{ b }+\frac{\sigma 4 \pi c ^2}{ c }=\frac{4 \pi \sigma\left[ a ^2- b ^2+ c ^2\right]}{ c }$
$c ( a - b + c )= a ^2- b ^2+ c ^2$
$c ( a - b )= a ^2- b ^2$
$c = a + b$
$c =5\,cm$
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