Two similar conducting balls having charges $+q$ and $-q$ are placed at a separation $d$ from each other in air. The radius of each ball is $r$ and the separation between their centres is $d(d > > r)$. Calculate the capacitance of the two ball system
Medium
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(b)
$V_A=\frac{k q}{r}+\frac{k(-q)}{d-r}$
$V_B=\frac{-k q}{r}+\frac{k q}{d-r}$
$V=V_A-V_B=\frac{2 k q}{r}-\frac{2 k q}{d-r}=\frac{2 q}{4 \pi \varepsilon_0}\left[\frac{1}{r}-\frac{1}{d-r}\right]$
$V=\frac{q}{2 \pi \varepsilon_0}\left[\frac{d-2 r}{(r)(d-r)}\right]$
$d \gg r$
$V=\frac{q}{2 \pi \varepsilon_0}$
$\frac{q}{V}=2 \pi \varepsilon_0 r=C$
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