Animated Solution for Physics - Electromagnetic Induction: An emf of $20\text{ V}$ is applied at time $t=0$ to a circuit containing in series $10\text{ mH}$ inductor and $5\Omega$ resistor. The ratio of the currents at time $t=\infty$ and at $t=40\text{ s}$ is close to (Take, $e^2=7.389$)

Solving the L-R Circuit Mystery

Imagine you are standing in front of a simple electrical circuit. You have a $20\text{ V}$ battery, a $5\Omega$ resistor, and a $10\text{ mH}$ inductor, all connected in series. The moment you close the switch, you might expect the current to instantly jump to its maximum value. But the inductor acts like a stubborn bouncer, opposing any sudden change in the flow of electrons.

Because of this opposition, the current doesn't spike instantly; instead, it grows gradually, following a beautiful exponential curve. Let's dive into the mathematics of this growth and uncover a fascinating anomaly hidden within this specific JEE problem.

The instantaneous current $I(t)$ during the growth phase in an L-R circuit is governed by the standard equation:

Here, $I_0$ represents the maximum, steady-state current that will flow through the circuit after an infinite amount of time ($t=\infty$). At this point, the inductor stops opposing the steady current, and the circuit behaves as if only the resistor is present. Therefore, $I(\infty )=I_0=\frac{E}{R}$.

The question asks for the ratio of the current at $t=\infty$ to the current at $t=40\text{ s}$. Let's set up our ratio:

Now, we need to carefully substitute our given values to evaluate the exponent term $\frac{R}{L}t$. We have $R=5\Omega$ and $L=10\text{ mH}=10\times 10^{-3}\text{ H}$.

We have arrived at a massive exponent of $20,000$. Let's plug this back into our ratio equation:

Mathematically, $e^{-20000}$ is an astronomically small number—so close to zero that it is practically indistinguishable from it. Therefore, the denominator becomes $1-0=1$, making our exact ratio exactly $1$.

However, if you look at the options provided—(a) $1.06$, (b) $1.15$, (c) $1.46$, (d) $0.84$—the number $1$ is nowhere to be found! In a high-pressure exam scenario, the standard strategy is to choose the nearest available option, which in this case is $1.06$.

Why would a JEE question have such a glaring mismatch? The secret lies in the hint provided at the end of the question: Take $e^2=7.389$.

Why would the examiner give you the value of $e^2$ if your exponent was going to be $20,000$? They wouldn't. This is a massive clue that the intended exponent was actually $2$.

If the exponent were $2$, the ratio would be:

This perfectly matches option (b)! For the exponent to be $2$, the time $t$ should have been $4\text{ ms}$ instead of $40\text{ s}$. This reveals that the question contained a typographical error. While the mathematically rigorous answer to the text as written leads us to approximate to $1.06$, understanding the setter's intent shows the beauty of reverse-engineering a physics problem.