The distance between charges $+q$ and $-q$ is $2l$ and between $+2q$ and $-2q$ is $4l$, all placed on a line with common centre $O$ as shown. The electrostatic potential at point $P$ at a distance $r$ ($r \gg l$) from centre $O$ is $-\alpha\dfrac{ql}{r^2}\times10^9\ \text{V}$, where the value of $\alpha$ is ______. (Use $\dfrac1{4\pi\varepsilon_0} = 9\times10^9\ \text{N m}^2\text{C}^{-2}$)
Numerical value type. Enter your answer.
Answer: 27
The pair $-q$ (at $-l$), $+q$ (at $+l$) is a dipole of moment $2ql$ pointing towards $+x$ (the $+q$ side).
The pair $+2q$ (at $-2l$), $-2q$ (at $+2l$) is a dipole of moment $2q\times4l = 8ql$ pointing towards $-x$.
Net dipole moment: $p = 6ql$ along $-x$. $OP$ makes $60^\circ$ with $+x$, i.e. $120^\circ$ with $\vec p$:
$$V = \frac{kp\cos120^\circ}{r^2} = -\frac{9\times10^9\times6ql\times\frac12}{r^2} = -27\frac{ql}{r^2}\times10^9\ \text{V}$$
$\alpha = 27$.
Solution by Sreeraj P, M.Sc Physics