Animated Solution for Physics - Electromagnetic Induction: A coil of inductance $300\text{ mH}$ and resistance $2\Omega$ is connected to a source of voltage $2\text{ V}$. The current reaches half of its steady state value in

Imagine you are pushing a heavy boulder. When you first apply force, it doesn't instantly jump to its maximum speed. It takes time to accelerate because of its mechanical inertia. Electrical circuits have their own version of inertia, and it's called inductance.

When you close a switch in a circuit containing a resistor ($R$) and an inductor ($L$) connected to a DC voltage source ($V$), the current doesn't instantaneously reach its maximum value. The inductor strongly opposes any sudden change in current by inducing a counter-electromotive force (EMF). This creates a fascinating "transient state" where the current gradually climbs up to its final, steady value.

The mathematics governing this gradual climb is one of the most elegant applications of differential equations in physics. The current $I(t)$ at any given time $t$ is described by the exponential growth equation:

Here, $I_0$ is the maximum, steady-state current that will eventually flow through the circuit after a very long time. According to Ohm's Law, once the inductor stops opposing the steady current, $I_0=\frac{V}{R}$. The term $\frac{L}{R}$ is known as the time constant ($\tau$) of the circuit, which dictates exactly how sluggish or snappy the circuit is.

Our mission is to find the exact moment when the current reaches exactly half of its maximum possible value. We don't need to know the final current to find this time; we just need to set our instantaneous current $I(t)$ to $\frac{I_0}{2}$.

Let's substitute this into our master equation:

Notice something beautiful? The $I_0$ term appears on both sides of the equation. We can completely cancel it out! This reveals a profound physical truth: the time it takes to reach a specific fraction of the maximum current is completely independent of the applied voltage.

Now, we rearrange the terms to isolate the exponential part. Moving the exponential to the left and the fraction to the right gives us:

To rescue our time variable $t$ from the exponent, we must take the natural logarithm ($\ln$) of both sides. Remember that $\ln \left(\frac{1}{2}\right)$ is the same as $-\ln 2$.

Isolating $t$, we get our final working formula:

We have successfully built our mathematical tool. Now, it's time to plug in the physical parameters provided in the problem:

Inductance ($L$): $300\text{ mH}=300\times 10^{-3}\text{ H}$ Resistance ($R$): $2\Omega$ Natural Log of 2 ($\ln 2$)*: $\approx 0.693$

Substituting these into our formula:

Rounding to the nearest sensible significant figure, we get $0.1\text{ s}$. The current will reach half of its maximum capacity in just a tenth of a second. The math is consistent, elegant, and perfectly describes the physical reality of the circuit!