question_answer

Figure shows a circular wire loop of radius r, carrying current i, placed in a perpendicular magnetic field B. The radius of cross-section of the wire is 'a'. Find decrease in the radius of the loop if the magnetic field is switched off.

[Young's modulus of the material of the wire is Y]

A) \[\frac{i{{r}^{2}}B}{2\pi Y{{a}^{2}}}\]

B) \[\frac{i{{r}^{2}}B}{\pi Y{{a}^{2}}}\]

C) \[\frac{i{{a}^{2}}B}{\pi {{r}^{2}}Y}\]

D) zero  

Correct Answer: B

Solution :

\[2T\sin \theta /2=Bi(r\theta )\] Or \[2T\left( \frac{\theta }{2} \right)=Bir\theta \]

\[\therefore \]      \[T=Bir\]

Stress, \[f=\frac{T}{\pi {{a}^{2}}}=\frac{Bir}{\pi {{a}^{2}}}\] Using, \[\frac{f}{e}=Y\] Or \[\left( \frac{Bir/\pi {{a}^{2}}}{\Delta r/r} \right)=Y\therefore \Delta r=\frac{i{{r}^{2}}B}{\pi Y{{a}^{2}}}\]