A dipole with dipole moment $p$ is placed at a distance $r$ from along charged wire with linear charge density $\lambda$. The force on the dipole is proportional to:

Solution

1. The electric field $E$ at a distance $r$ from along charged wire with linear charge density $\lambda$ is given by $E=\frac{\lambda }{2\pi {ϵ}_{0}r}$.
2. The force $F$ exerted on a dipole with moment $p$ in an electric field is $F=p\frac{dE}{dr}$.
3. Differentiating the expression for the electric field with respect to $r$ gives $\frac{dE}{dr}=-\frac{\lambda }{2\pi {ϵ}_0{r}^2}$, which implies $\frac{dE}{dr}\propto \frac{\lambda }{r^2}$.
4. Substituting this into the force equation, we find $F\propto p\cdot \frac{\lambda }{r^2}$.
5. Hence the answer is (B).