Animated Solution for Physics - Electromagnetic Induction: A coil of inductance $8.4\text{ mH}$ and resistance $6\Omega$ is connected to a $12\text{ V}$ battery. The current in the coil is $1\text{ A}$ at approximately the time

Imagine you are flipping a switch to turn on a circuit containing a battery, a resistor, and an inductor. Unlike a simple lightbulb circuit where the current instantly reaches its maximum, an inductor adds a layer of "inertia" to the flow of electricity.

According to Lenz's Law, the inductor opposes any sudden change in current by inducing a back EMF. This means the current has to fight its way up, growing gradually over time. In our specific problem, we have a $12\text{ V}$ battery, a $6\Omega$ resistor, and an $8.4\text{ mH}$ inductor. We need to find exactly when the current reaches $1\text{ A}$.

To track this gradual climb, we use the standard equation for the growth of current in an L-R circuit:

Here, $i_0$ represents the maximum steady-state current, and ${\tau }_L$ is the time constant of the circuit. Let's break these down.

First, what is the maximum current $i_0$? After a long time, the current stops changing, meaning the inductor stops fighting and acts just like a regular wire. The circuit behaves purely resistively. Using Ohm's Law:

Next, we calculate the time constant ${\tau }_L$, which dictates how "sluggish" the circuit is. A larger inductance makes it slower, while a larger resistance makes it faster:

Now we have all the pieces of the puzzle. We want to find the time $t$ when the current $i(t)$ is exactly $1\text{ A}$. Let's substitute our known values into the master equation:

Dividing both sides by 2 gives:

Rearranging the terms to isolate the exponential part, we get:

To bring the time variable $t$ down from the exponent, we take the natural logarithm ($\ln$) of both sides. Remember that $\ln \left(\frac{1}{2}\right)=-\ln (2)$:

The negative signs beautifully cancel out. Now, we just multiply:

Using the standard approximation $\ln (2)\approx 0.693$:

This is $0.97\text{ ms}$, which is approximately $1\text{ ms}$. The math perfectly aligns with option (d).

I know exponential equations can sometimes look intimidating, but by breaking them down into steady-state current and time constant, the physics naturally guides the algebra!