Physics - Electrostatics Question with Solution | TestHub

With $\lambda ={\lambda }_0\sin \theta$ on ring A and $\lambda ={\lambda }_0\cos \theta$ on ring B, each ring acts as a dipole of moment $p=\pi {\lambda }_0{R}^2$: A along $\hat{y}$ and B along $\hat{x}$, while the line of centres is along x.

Interaction energy $U=\frac{k}{r^3}[{\vec{p}}_A\cdot {\vec{p}}_B-3({\vec{p}}_A\cdot \hat{r})({\vec{p}}_B\cdot \hat{r})]=0$ since ${\vec{p}}_A\perp {\vec{p}}_B$ and ${\vec{p}}_A\perp \hat{r}$ - (D) correct.

On the x-axis the field of A is $E_A=-\frac{k{p}_A}{r^3}$, so the force on B is $F=p_B\left|\frac{d{E}_A}{dr}\right|=\frac{3k{p}_A{p}_B}{r^4}=\frac{3}{4}\frac{{\lambda }_0^2\pi }{{ϵ}_0}{\left(\frac{R}{r}\right)}^4$ - (A) correct.