Animated Solution for Physics - Electromagnetic Induction: An inductor of $10\text{ mH}$ is connected to a $20\text{ V}$ battery through a resistor of $10\text{ k}\Omega$ and a switch. After a long time, when maximum current is set up in the circuit, the current is switched off. The current in the circuit after $1\mu \text{s}$ is $\frac{x}{100}\text{ mA}$. Then, $x$ is equal to ...... (Take, $e^{-1}=0.37$)

The phenomenon of current decay in an $L-R$ circuit is one of the most fascinating displays of electromagnetic inertia. It perfectly encapsulates Lenz's Law and the fundamental nature of inductors. Let's embark on a detailed journey to understand exactly what happens when we flip that switch.

Before we dive into the mathematics, we must understand our primary actor: the inductor. An inductor is essentially a coil of wire, but its behavior is anything but simple. When current flows through it, it generates a magnetic field. This magnetic field stores energy.

However, nature abhors sudden changes. If you try to change the current flowing through an inductor, the magnetic field changes, which in turn induces an electromotive force (EMF) that opposes the change. This is the essence of Lenz's Law. The inductor acts as an electromagnetic shock absorber, smoothing out any sudden spikes or drops in current.

The Calm Before the Storm

In our problem, the inductor ($L=10\text{ mH}$) and the resistor ($R=10\text{ k}\Omega$) are initially connected to a $20\text{ V}$ battery for a "long time." This phrase is a classic physics code word for steady state.

In a DC circuit, once the steady state is reached, the current is no longer changing. Since the induced EMF in an inductor is given by $V_L=L\frac{dI}{dt}$, a constant current ($\frac{dI}{dt}=0$) means the inductor has zero voltage drop across it. It behaves exactly like a plain, ideal wire!

Therefore, the maximum current $I_0$ flowing through the circuit is limited solely by the resistor. Using Ohm's Law, we can easily calculate this:

Substituting our values:

This $2\text{ mA}$ is the peak current, and the inductor is happily storing energy in its magnetic field, completely unaware of the impending disruption.

The Magnetic Collapse

Suddenly, the battery is switched off, and the circuit is shorted to form a closed loop containing only the inductor and the resistor. If this were a simple resistor circuit, the current would instantly drop to zero. But we have an inductor!

As the current tries to drop to zero, the magnetic field inside the inductor begins to collapse. This collapsing field induces a strong EMF that pushes the current to keep flowing in the same direction. The inductor temporarily becomes the power source, driving current through the resistor.

However, as the current flows through the resistor, energy is dissipated as heat. With less energy available, the magnetic field weakens further, the induced EMF drops, and the current gradually decreases. This is a classic exponential decay.

The mathematical model for this decaying current is beautifully elegant:

Here, $\tau$ is the time constant of the circuit, defined as $\tau =\frac{L}{R}$. It represents the time it takes for the current to decay to approximately $37\%$ ($1/e$) of its initial value. Let's calculate our specific time constant:

Notice how the problem setter deliberately chose the time $t=1\mu \text{s}$ to perfectly match our time constant! This is a common trick in competitive exams to simplify the final calculation.

Now, we substitute all our known values into the decay equation to find the current at $t=1\mu \text{s}$:

The problem kindly provides the value of $e^{-1}=0.37$. Substituting this in:

The final step is simply matching our result to the format requested by the question, which is $\frac{x}{100}\text{ mA}$.

Thus, we can clearly see that $x=74$.

This problem is a masterful blend of conceptual understanding and precise execution. By visualizing the physical reality of the collapsing magnetic field and carefully managing our units, we transformed a potentially intimidating differential equation into a straightforward algebraic substitution.