Electro Magnetic Induction And Alternating Currents Question 455
Question: Magnetic flux linked with a stationary loop of resistance R varies with respect to time during the time period T as follows: $ \phi =at(T-t) $ The amount of heat generated in the loop during that time (inductance of the coil is negligible) is
Options:
A) $ \frac{aT}{3R} $
B) $ \frac{a^{2}T^{2}}{3R} $
C) $ \frac{a^{2}T^{2}}{R} $
D) $ \frac{a^{2}T^{3}}{3R} $
Show Answer
Answer:
Correct Answer: D
Solution:
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Given that $ \phi =at(T-t) $
Induced emf, $ E=\frac{d\phi }{dt}=\frac{d}{dt}[at(T-t)] $ = at $ (0-1)+a(T-t)=a(T-2t) $
So, induced emf is also a function of time.
$ \therefore $ Heat generated in time T is $ H\int\limits_{0}^{T}{\frac{E^{2}}{R}}dt=\frac{a^{2}}{R}\int\limits_{0}^{T}{{{(T-2t)}^{2}}} $
$ dt=\frac{a^{2}T^{3}}{3R} $
