question_answer

A flexible wire loop in the shape of a circle has radius that grown linearly with time. There is a magnetic field perpendicular to the plane of the loop that has a magnitude inversely proportional to the distance from the center of the loop, \[B(r)\propto \frac{1}{r}\].How does the emf E vary with time?

A) \[E\propto {{t}^{2}}\]  

B) \[E\propto t\]

C) \[E\propto \sqrt{t}\]        

D) E is constant

Correct Answer: D

Solution :

[d] Let radius of the loop is r at any time t and in further time dt, radius increases by dr. The change in flux: \[d\phi =(2\pi rdr)B\] \[\Rightarrow \,\,e=\frac{d\phi }{dt}=2\pi r\left( \frac{dr}{dt} \right)\frac{k}{r}\] \[\Rightarrow \,\,\,e=2\pi ck\] (constant) \[\left[ \because \,\,\frac{dr}{dt}=c,\,B=\frac{k}{r} \right]\] The change in flux:                \[d\phi =(2\pi rdr)B\]