Electro Magnetic Induction And Alternating Currents Question 319
Question: A coil of wire having finite inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time t = 0, so that a time-dependent current $ I_{1}(t) $ starts flowing through the coil. If $ I_{2}(t) $ is the current induced in the ring. and $ B(t) $ is the magnetic field at the axis of the coil due to $ I_{1}(t), $ then as a function of time (t > 0), the product I2 (t) B(t) [IIT-JEE (Screening) 2000]
Options:
A) Increases with time
B) Decreases with time
C) Does not vary with time
D) Passes through a maximum
Show Answer
Answer:
Correct Answer: D
Solution:
Using k1, k2 etc, as different constants.
$ I_{1}(t)=k_{1}[1-{{e}^{-t/\tau }}],\ B(t)=k_{2}I_{1}(t) $
$ I_{2}(t)=k_{3}\frac{dB(t)}{dt}=k_{4}{{e}^{-t/\tau }} $
$ \therefore I_{2}(t)\ B(t)=k_{5}[1-{{e}^{-t/\tau }}][{{e}^{-t/\tau }}] $
This quantity is zero for $ t=0 $ and $ t=\infty $ and positive for other value of t. It must, therefore, pass through a maximum.
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