Question

A square wire frame of linear mass density $\lambda =1kg/m$, side length L =1M, carrying a current of i =2A and with its plane in an X Y plane as shown. An uniform magnetic field B=2T directed along Y axis is switched on, then find the instantaneous angular acceleration $\left(rad/se{c}^2\right)$ of the frame.___

Answer 1

$\tau =MB{I}_x\alpha =I{L}^{2}B{I}_x=2\left(\frac{\left(\lambda L\right)}{12}\right(L^2\left)\right)+2\left(\lambda L\right){\left(\frac{L}{2}\right)}^2=\frac{2\lambda L^3}{12}+\frac{2\lambda L^3}{4}=\frac{\lambda L^3}{2}[\frac{1}{3}+1]=\frac{2\lambda L^3}{3}\frac{2\lambda L^3}{3}\alpha =I{L}^{2}B\alpha =\frac{3BI}{2\lambda L}=\frac{3\times 2\times 2}{2\times 1\times 1}=6rad/s^2$