An electric dipole is placed on $x$-axis in proximity to a line charge of linear charge density $3.0\times10^{-6}\ \text{C m}^{-1}$. Line charge is placed on $z$-axis and positive and negative charge of dipole is at a distance of 10 mm and 12 mm from the origin respectively. If total force of 4 N is exerted on the dipole, find out the amount of positive or negative charge of the dipole.
Answer: (D) 4.44 $\mu$C
Field of the line charge: $E = \dfrac{2k\lambda}{r}$ with $2k\lambda = 2\times9\times10^9\times3\times10^{-6} = 5.4\times10^4\ \text{V}$.
The $+q$ at 10 mm is pushed away and the $-q$ at 12 mm is pulled in; the net force is
$$F = 2k\lambda q\left(\frac{1}{0.010} - \frac{1}{0.012}\right) = 5.4\times10^4\times16.67\,q = 9\times10^5\,q$$
$9\times10^5\,q = 4 \Rightarrow q \approx 4.44\times10^{-6}\ \text{C} = 4.44\ \mu\text{C}$.
Solution by Sreeraj P, M.Sc Physics