JEE Main · 2024 · 31 Jan 2024 · Shift 2 · Electrostatics

The +q, −q pair (separation 2l) forms a dipole of moment $p_1=q(2l)=2ql$. The +2q, −2q pair (separation 4l) forms a dipole of moment $p_2=2q(4l)=8ql$, oriented opposite to $p_1$ along the same line. The net dipole moment of the arrangement is

$p_{net}=p_2-p_1=6ql$

directed along the axis toward the +2q side.

Point P lies such that the dipole axis (measured from the +2q direction) makes an angle of 120° with OP. The potential of a dipole at a field point is

$V=\frac{1}{4\pi {ϵ}_0}\frac{p\cos \theta }{r^2}$

$V=9\times 10^9\times \frac{6ql\cos 120^{\circ }}{r^2}=9\times 10^9\times 6ql\times \left(-\frac{1}{2}\right)\frac{1}{r^2}=-27\times 10^9\frac{ql}{r^2}\text{V}$

Comparing with $-\alpha \left[\frac{ql}{r^2}\right]\times 10^9$, $\alpha =27$