Electro Magnetic Induction And Alternating Currents Question 461
Question: 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?
Options:
A) $ E\propto t^{2} $
B) $ E\propto t $
C) $ E\propto \sqrt{t} $
D) E is constant
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Answer:
Correct Answer: D
Solution:
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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( \frac{dr}{dt} )\frac{k}{r} $
$ \Rightarrow ,e=2\pi ck $ (constant) $ [ \because \frac{dr}{dt}=c,,B=\frac{k}{r} ] $The change in flux: $ d\phi =(2\pi rdr)B $
