Question

A circular loop of radius R is bent along a diameter and given a shape as shown in the figure. One of the semicircles (KNM) lies in the figure. One of the semicircles (KNM) lies in the x-z plane and the other one (KLM) in the y-z plane with their centres at the origin. Current I is flowing through each of the semicircles as shown in figure.
(a) A particles of charge q is released at the origin with a velocity $v=-v0\hat{i}$. Find the instantaneous force f on the particle. Assume that space is gravity free.
(b) If an external uniform magnetic field $B\hat{j}$ is applied, determined the forces $F_1$ and $F_2$ on the semicircles LKM and KMN due to this field and the net force F on the loop. [Magnetic field at the origin ${\overrightarrow{B}}_{KMN}+{\overrightarrow{B}}_{KLM}=\frac{{\mu }_{0}I}{4R}(\hat{j}-\hat{i})]$

Answer 1

(a) Required force, $\overrightarrow{F}=q\left[\right(-v_0\hat{i})\times \frac{{\mu }_{0}I}{4R}(\hat{i}-\hat{j}\left)\right]=\frac{{\mu }_{0}I{v}_{0}q}{4R}[-\hat{k}]$

(b) Force on semicircular loop KLM due to external field $B\hat{j}$ is same as the force experienced by straight current carrying conductor along KM = $-I\left(2R\hat{k}\right)\times B\hat{j}=2IBR\hat{j}$

Similarly force on the semicircular loop $KMN=2IBR\hat{i}$

$\therefore$ Net force on the loop = $\overrightarrow{F}=4IBR\hat{i}$.