First slide

Inductance

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 I₁(t) starts flowing through the coil. If l₂(t) is the current induced in the ring and B(t) is the magnetic field at the axis of the coil due to l₁(t), then as a function of time (t > 0), the product I₂(t) B(t)

Difficult

A

Increases with time
B

Decreases with time
C

Does not vary with time
D

Passes through a maximum

Solution

$Using k_{1}, k_{2} etc, as different constants.$
$\begin{array}{l} I_{1} (t) = k_{1} [1 - e^{- t / τ}], B (t) = k_{2} I_{1} (t) \\ I_{2} (t) = k_{3} \frac{d B (t)}{d t} = k_{4} e^{- t / τ} \\ ∴ I_{2} (t) B (t) = k_{5} [1 - e^{- t / τ}] [e^{- t / τ}] \end{array}$
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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