A circular conducting coil of radius 1 m is being heated by the change of magnetic field B passing p

Q: Why is the induced EMF constant in this problem?

Because the magnetic field decreases linearly with time. When you differentiate a linear function, you get a constant. A constant rate of change of flux means a constant induced EMF.

Q: Why didn't we have to integrate to find the total energy?

Integration is necessary when the power varies with time. Here, because the EMF is constant and the resistance is constant, the power dissipated is also constant. So, we can simply multiply power by total time.

Q: What happens to the negative sign in Faraday's Law?

The negative sign represents Lenz's Law, indicating the direction of the induced EMF (it opposes the change in flux). However, when calculating power ($\frac{\varepsilon^2}{R}$), the EMF is squared, so the negative sign becomes irrelevant for the energy calculation.

Q: How did we know to set the magnetic field to zero to find the time?

The problem asks for the energy dissipated 'before the magnetic field is switched off completely'. 'Switched off completely' physically translates to the magnetic field magnitude $B$ becoming exactly zero.

Q: Why did we convert the resistance to ohms?

Standard SI units are crucial for energy to come out in Joules. The resistance was given in micro-ohms ($\mu\Omega$), so we multiplied by $10^{-6}$ to convert it to Ohms ($\Omega$) before plugging it into the power formula.

Question

A circular conducting coil of radius $1	ext{ m}$ is being heated by the change of magnetic field $\mathbf{B}$ passing perpendicular to the plane in which the coil is laid. The resistance of the coil is $2\mu\Omega$. The magnetic field is slowly switched off such that its magnitude changes in time as
$$B = \frac{4}{\pi} 	imes 10^{-3} \left(1 - \frac{t}{100}ight)	ext{T}$$
The energy dissipated by the coil before the magnetic field is switched off completely is $E = \dots\dots 	ext{mJ}$.

Sigma Percentile · Accepted Answer

### Visualizing the Setup

Imagine a circular conducting coil resting peacefully on a table. A uniform magnetic field $\mathbf{B}$ is passing straight through it, perpendicular to the plane of the coil. However, this magnetic field is not static; it is slowly dying out.

We are given the radius of the coil, $r = 1	ext{ m}$, and its resistance, $R = 2\mu\Omega = 2 	imes 10^{-6} \Omega$. The magnitude of the magnetic field is decreasing according to the linear equation:

$$B(t) = \frac{4}{\pi} 	imes 10^{-3} \left(1 - \frac{t}{100}ight)	ext{T}$$

Our mission is to find the total heat energy dissipated by the coil before the magnetic field completely vanishes.

### The Core Principle: Faraday's Law

Whenever the magnetic flux through a closed loop changes, an electromotive force (EMF) is induced. This is the heart of Faraday's Law of Electromagnetic Induction. The induced EMF $\varepsilon$ is given by the negative rate of change of magnetic flux $\phi$:

$$\varepsilon = -\frac{d\phi}{dt}$$

Since the magnetic field is perpendicular to the coil's area, the flux is simply $\phi = B \cdot A$. The area of our circular coil is constant ($A = \pi r^2$), so we can pull it out of the derivative:

$$\varepsilon = -A \frac{dB}{dt}$$

### Calculating the Rate of Change

To find the induced EMF, we first need to determine how fast the magnetic field is changing. We do this by differentiating the given magnetic field equation with respect to time $t$:

$$\frac{dB}{dt} = \frac{d}{dt} \left[ \frac{4}{\pi} 	imes 10^{-3} \left(1 - \frac{t}{100}ight) ight]$$

Expanding the bracket mentally, the constant term $\frac{4}{\pi} 	imes 10^{-3}$ differentiates to zero. We are left with:

$$\frac{dB}{dt} = \frac{4}{\pi} 	imes 10^{-3} 	imes \left(-\frac{1}{100}ight) = -\frac{4}{\pi} 	imes 10^{-5} 	ext{ T/s}$$

Notice that the rate of change is a **constant negative value**, meaning the field is decreasing steadily.

### The Beauty of Cancellation: Finding the EMF

Now, let's substitute this rate of change back into our EMF equation. The area of the coil is $A = \pi (1)^2 = \pi 	ext{ m}^2$.

$$\varepsilon = -(\pi) \left( -\frac{4}{\pi} 	imes 10^{-5} ight)$$

Look at how beautifully the math works out! The $\pi$ in the area cancels perfectly with the $\pi$ in the denominator of the rate of change. The negative signs also multiply to give a positive value. We are left with a beautifully simple, constant induced EMF:

$$\varepsilon = 4 	imes 10^{-5} 	ext{ V}$$

### Determining the Time Frame

To calculate the total energy dissipated, we need to know how long this current flows. The problem states we need the energy dissipated "before the magnetic field is switched off completely." This physically translates to the moment when $B = 0$.

Let's set our magnetic field equation to zero and solve for $t$:

$$\frac{4}{\pi} 	imes 10^{-3} \left(1 - \frac{t}{100}ight) = 0$$

$$1 - \frac{t}{100} = 0 \implies t = 100 	ext{ s}$$

So, the induction process lasts for exactly 100 seconds.

### The Final Energy Calculation

Electrical power dissipated in a resistor is given by $P = \frac{\varepsilon^2}{R}$. Because our induced EMF is constant, the power dissipated is also constant. Therefore, the total energy $E$ is simply power multiplied by the total time:

$$E = P 	imes t = \frac{\varepsilon^2}{R} 	imes t$$

Let's plug in our values. Be careful to use the resistance in Ohms ($2 	imes 10^{-6} \Omega$):

$$E = \frac{(4 	imes 10^{-5})^2}{2 	imes 10^{-6}} 	imes 100$$

$$E = \frac{16 	imes 10^{-10}}{2 	imes 10^{-6}} 	imes 100$$

$$E = 8 	imes 10^{-4} 	imes 100 = 80 	imes 10^{-3} 	ext{ J}$$

Since $10^{-3} 	ext{ J}$ is exactly one millijoule (mJ), our final answer is:

**$$E = 80 	ext{ mJ}$$**

This problem is a fantastic demonstration of how linear changes in a magnetic field lead to constant induced voltages, making energy calculations wonderfully straightforward!

Enter Numerical Value:

Visualized Solution

Visualizing the Setup

Faraday's Law of Induction

Rate of Change of Magnetic Field

Calculating Induced EMF

Time to Switch Off

Energy Dissipated Formula

Final Calculation

The Way Forward

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

Visualizing the Setup

The Core Principle

Calculating the Rate of Change

The Beauty of Cancellation

Determining the Time Frame

The Final Energy Calculation

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Enter Numerical Value:

Visualized Solution

Visualizing the Setup

Faraday's Law of Induction

Rate of Change of Magnetic Field

Calculating Induced EMF

Time to Switch Off

Energy Dissipated Formula

Final Calculation

The Way Forward

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

Visualizing the Setup

The Core Principle

Calculating the Rate of Change

The Beauty of Cancellation

Determining the Time Frame

The Final Energy Calculation

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