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Three Charged Spheres

Let's discuss a problem where we will find the charge of the sphere with the help of the three charged spheres problem. Try before reading the solution.

The Problem:

Consider three concentric metallic spheres \(A\), \(B\) and \(C\) of radii of \(a\), \(b\), \(c\) respectively where a<b<c . \(A\) and \(B\) are connected whereas C is grounded The potential of the middle sphere \(B\) is raised to \(V\) then what is the charge on the sphere \(C\)?

Solution:

Three concentric metallic spheres \(A\), \(B\) and \(C\) have radii of \(a\), \(b\), \(c\) respectively where a<b<c . \(A\) and \(B\) are connected whereas C is grounded The potential of the middle sphere \(B\) is raised to \(V\).

$$ V=\frac{Kq}{b}+\frac{KQ}{c}$$ $$
\Rightarrow \frac{k(q+Q)}{c}=0$$
$$\Rightarrow q+Q=0$$
$$\Rightarrow q=-Q
$$
$$ \frac{k(-Q)}{b}+\frac{KQ}{c}=V$$
$$\Rightarrow KQ(\frac{1}{c}-\frac{1}{b})=V$$
$$ Q=\frac{bcV}{(b-c)}4\pi\epsilon_0$$

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