A coil of wire having inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time , so that a time-dependent current starts flowing through the coil. If is the current induced in the ring, and

Question

A coil of wire having 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 $I_{2} (t) B (t)$ [2000]

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(2)

A conducting ring is placed coaxially within a coil and the coil has inductance as well as resistance. The magnetic field at the centre of the coil $B (t) = μ_{0} n I_{1}$ .

As the current increases, $B$ will also increase with time till it reaches a maximum value (when the current becomes steady).

Induced emf in the ring

$e = - \frac{d ϕ}{d t} = - \frac{d}{d t} (\vec{B} \cdot \vec{A}) = - A \frac{d}{d t} (μ_{0} n I_{1})$

$∴$ Induced current in the ring

$I_{2} (t) = \frac{| e |}{R} = \frac{μ_{0} n A}{R} \frac{d I_{1}}{d t}$

[ $\frac{d I_{1}}{d t}$ decreases with time and hence $I_{2}$ also decreases with time.]

where $I_{1} = I_{\max} (1 - e^{- t / τ})$

The relevant graphs are as follows.

Hence, as a function of time $(t > 0)$ , the product $I_{2} (t) B (t)$ decreases with time.

Solution

1 increases with time

2 decreases with time

3 does not vary with time

4 passes through a maximum

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