Three charged particles having charges $q,-2 q$ and $q$ are placed in a line at points $(-a, 0),(0,0)$ and $(a, 0)$ respectively. The expression for electric potential at $P(r, 0)$ for $r \gg a$ is ...............
Diffcult
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(b)
$E_1=\frac{2 k p}{\left(r+\frac{a}{2}\right)^3} \quad E_2=\frac{2 k p}{\left(r-\frac{a}{2}\right)^3}$
$E_2-E_1=\frac{2 k p}{\left(r-\frac{a}{2}\right)^3}-\frac{2 k p}{\left(r+\frac{a}{2}\right)^3}$
$\frac{2 k p}{r^3}\left[\frac{1}{\left(1-\frac{a}{2 r}\right)^3}-\frac{1}{\left(1+\frac{a}{2 r}\right)^3}\right]$
$\frac{2 k p}{r^3}\left[1+\frac{3 a}{2 r}-\left(1-\frac{3 a}{2 r}\right)\right]$
$\frac{2 k p}{r^3}\left[\frac{3 a}{r}\right]$
$E=\frac{6 k p a}{r^4}=\frac{{ }^3 k}{2 k \pi \varepsilon_0} \frac{p a}{r^4}=\frac{3 p a}{2 \pi \varepsilon_0 r^4}$
$V=\frac{-k p}{\left(r+\frac{a}{2}\right)^2}+\frac{k p}{\left(r-\frac{a}{2}\right)^2}=\frac{k p}{r^2}\left[1+\frac{2 a}{2 r}-1+\frac{2 a}{2 r}\right]=\frac{2 k p a}{r^3}$
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