Question

A rectangular loop of wire is oriented with the left corner at the origin, one edge along X-axis and the other edge along Y-axis as shown in the figure. A magnetic field is into the page and has a magnitude that is given by $\beta =\alpha y$ where $\alpha$ is constant. Find the total magnetic force on the loop if it carries current $i$.

Answer 1

F = F₁ + F₂ + F₃ + F₄

$i\int (d{l}_1\times B_1)+i\int ({dl}_2\times B_2)+i\int ({dl}_3\times B_3)+i\int ({dl}_4\times B_4)$

$\left[i{\int }_0^{a}dy\hat{j}\times \left(\alpha y\right)(-\hat{k})\right]+i(a\hat{i}\times \alpha y(-\hat{k}))+i{\int }_0^{a}d\widehat{yj}\times \left(\alpha y\right)\hat{k}+I\times 0$

$F_1=-i\alpha \frac{a^2}{2}\hat{i}+i\alpha a^2\hat{j}+i\frac{\alpha a^2}{2}\hat{i}=i\alpha a^2\hat{j}$