Question

A coil of wire having finite inductance and resistance has a conducting ring placed coaxially within it. The coil is connected 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 dut to $I_1(t)$ then as a function of time (t > 0), the product $I_2(t) \,B(t)$

• A. increases with time
• B. decreases with time
• C. does not vary with time
• D. passes through a maximum

Answer 1

Answer: (D)

passes through a maximum

Answer 2

The equation of $I_1 (t) , I_2 (t)$ and $B(t)$ will take the following forms $I_1 (t) = K_1 (1 - e^{-k \, 2t} ) \rightarrow$ current growth in L - R circuit $B(t) = K_3 (1 - e^{-k \, 2t} ) \rightarrow B(t) \propto I_1 (t)$ $I_2 (t) = K_4e^{-k\, 2t}$ $\left[ I_2 (t) = \frac{e_2}{R} \, and \, e_2 \, \propto = - M \frac{dI_1}{dt}\right]$ Therefore, the product $I_2 (t) B (t) = K_5 e^{-k \, 2t} ( 1 - e^{-k \, 2t})$ The value of this product zero at $t = 0$ and $t = 0$ . Therefore, the product will pass throiigh a maximum value.