Animated Solution for Physics - Electromagnetic Induction: An inductor coil stores 64 J of magnetic field energy and dissipates energy at the rate of 640 W when a current of 8A is passed through it. If this coil is joined across an ideal battery, find the time constant of the circuit in seconds.

Welcome to a fascinating journey into the heart of electromagnetic circuits! Today, we are going to dissect a classic problem that beautifully bridges the gap between ideal theoretical components and real-world electrical devices.

Imagine you are an electrical engineer holding a physical inductor coil in your hand. In textbook diagrams, we often treat inductors as magical, pure components that only store magnetic energy. But in reality, that coil is made of a long, tightly wound copper wire. And as we know, every real wire has some electrical resistance.

This duality is the core of our problem. To analyze this "real" coil mathematically, we must split its personality into two ideal components connected in series: an ideal inductor $L$ and an ideal resistor $R$.

When a steady current $i$ flows through our real coil, two distinct physical phenomena occur simultaneously.

First, the current generates a magnetic field that threads through the loops of the coil. This magnetic field acts as a reservoir of energy. The ideal inductor part of our model is responsible for this energy storage.

Second, as the electrons march through the copper wire, they collide with the vibrating atoms of the metal lattice. These collisions generate heat, which is radiated away into the environment. The ideal resistor part of our model is responsible for this power dissipation.

By understanding this separation of duties, we can tackle the given data with absolute clarity.

The problem states that the coil stores $64\text{ J}$ of magnetic field energy when a steady current of $8\text{ A}$ is passed through it.

We know that the energy $U_m$ stored in an ideal inductor is given by the elegant equation:

This equation tells us that the energy scales linearly with the inductance $L$, but quadratically with the current $i$. Let's substitute our known values into this formula:

Squaring the current gives us $64$.

Notice how beautifully the numbers align! The $64$ on both sides cancels out perfectly.

Multiplying both sides by $2$, we find the inductance:

We have successfully extracted the first hidden parameter of our real coil. The pure inductance is exactly $2\text{ Henrys}$.

Next, we turn our attention to the heat being generated. The problem tells us that the coil dissipates energy at a rate of $640\text{ W}$.

Remember, our ideal inductor $L$ is completely lossless. All of this $640\text{ W}$ of power is being burned off by the internal resistance $R$. The formula for Ohmic power dissipation is:

Once again, we substitute our steady current of $8\text{ A}$ and the given power:

To isolate $R$, we simply divide both sides by $64$:

And just like that, the second hidden parameter is revealed. The internal resistance of the copper wire is $10\text{ Ohms}$.

Now comes the grand finale. The question asks for the time constant of the circuit if this coil is joined across an ideal battery.

But what exactly is a time constant?

When you connect an inductor to a battery, the current doesn't instantly jump to its maximum value. The inductor fights back! It creates a back-EMF that opposes the sudden change in current. This creates a "sluggishness" or electrical inertia in the circuit.

The time constant, denoted by the Greek letter $\tau$ (tau), is the mathematical measure of this sluggishness. It represents the time it takes for the current to reach approximately $63.2\%$ of its final steady-state value.

For a series $L-R$ circuit, the time constant is simply the ratio of the inductance to the resistance:

We have already done the hard work of finding $L$ and $R$. All that's left is to plug them in:

The time constant of our circuit is $0.2\text{ seconds}$.

This problem is a masterclass in component modeling. By breaking down a complex, real-world object into its fundamental ideal properties, we were able to use basic energy and power formulas to reveal its hidden characteristics.

Always remember: in physics, the real world is just a combination of ideal concepts waiting to be deciphered!