Physics - Magnetism Question with Solution | TestHub

The current element is $d\vec{l}=dx\hat{i}$. The magnetic force on this element is $d\vec{F}=I(d\vec{l}\times \vec{B})=I(dx\hat{i}\times B_{0}x\hat{k})=I{B}_{0}x(\hat{i}\times \hat{k})dx=I{B}_{0}x(-\hat{j})dx$. Integrating from $x=0$ to $x=L$, the total force is $\vec{F}={\int }_0^L-I{B}_{0}x\hat{j}dx=-I{B}_0\hat{j}[\frac{x^2}{2}{]}_0^L=-\frac{1}{2}I{B}_0{L}^2\hat{j}$. So A and B are incorrect. If the current flows from $x=L$ to $x=0$, $d\vec{l}=-dx\hat{i}$, which reverses the direction of $d\vec{F}$ and thus the total force. So C is correct. The magnitude of the force is $\frac{1}{2}I{B}_0{L}^2$, which is proportional to $L^2$. So D is correct.