Problem 72 - Physics for Scientists and Engineers with Modern Physics

Question

What is the energy dissipated as a function of time in a circular loop of 18 turns of wire having a radius of \(10.0 \mathrm{~cm}\) and a resistance of \(2.0 \Omega\) if the plane of the loop is perpendicular to a magnetic field given by $$ B(t)=B_{0} e^{-t / \tau} $$

Step-by-Step Solution

Answer

The energy dissipated as a function of time is expressed by \( E(t) = \int_{0}^{t} P(t') \, dt' \) using the derived power function.

Step 1: Understanding the Problem

We need to calculate the energy dissipated over time in a circular wire loop subject to a changing magnetic field. The magnetic field is expressed as \( B(t) = B_0 e^{-t / \tau} \). We have 18 turns, each with a radius of 10 cm, and the loop has a resistance of 2 \( \Omega \).

Step 2: Calculate the Magnetic Flux

The magnetic flux \( \Phi(t) \) through one loop is \( \Phi(t) = B(t) \cdot A \), where \( A = \pi r^2 \) is the area of the loop. Substituting the area, we get \( A = \pi \times (0.1)^2 = 0.01\pi \) sq. meters. Therefore, \( \Phi(t) = B_0 e^{-t/\tau} \times 0.01\pi \).

Step 3: Find the Induced EMF

Using Faraday's law of electromagnetic induction, the induced EMF \( \mathcal{E} \) is given by \( \mathcal{E} = -N \frac{d\Phi}{dt} \). Calculating the derivative of \( \Phi(t) \), we have \( \frac{d\Phi}{dt} = -B_0 e^{-t/\tau} \frac{1}{\tau} \times 0.01\pi \). Thus, \( \mathcal{E} = 18 \times (-B_0 e^{-t/\tau} \frac{1}{\tau} \times 0.01\pi) \).

Step 4: Determine the Current in the Loop

The current \( I(t) \) in the circuit is given by Ohm's law: \( I(t) = \frac{\mathcal{E}}{R} \). Substitute \( \mathcal{E} \) and \( R = 2 \Omega \), to get \( I(t) = \frac{18 \times (-B_0 e^{-t/\tau} \frac{1}{\tau} \times 0.01\pi)}{2} \).

Step 5: Compute the Power Dissipated

The power dissipated in the circuit as a function of time is \( P(t) = I^2(t) R \). Substitute \( I(t) \) to express power in terms of time: \( P(t) = \left(\frac{18 \times (-B_0 e^{-t/\tau} \frac{1}{\tau} \times 0.01\pi)}{2}\right)^2 \times 2 \).

Step 6: Express Energy as a Function of Time

The energy \( E(t) \) dissipated as a function of time is the integral of the power from time 0 to \( t \), which is \( E(t) = \int_{0}^{t} P(t') \, dt' \). Substitute the expression for \( P(t) \) and integrate to find \( E(t) \) in terms of known quantities.

Key Concepts

Magnetic Flux

Magnetic flux is a foundational concept in the study of electromagnetism. It describes the amount of magnetic field passing through a certain area. In simple terms, it's like sunlight streaming through a window; the wider the window, the more light that comes in. For our loop of wire, think of it as how much of the magnetic field goes through the circle made by the wire.

To calculate magnetic flux for a loop, we use the formula: \( \Phi(t) = B(t) \cdot A \). Here, \( B(t) \) is the magnetic field at a certain time, and \( A \) is the area the field is passing through. In this problem, the circular loop has a radius, so the area \( A \) is \( \pi r^2 \), where \( r = 0.1 \) m. Hence, \( A = 0.01 \pi \) square meters.

• Formula: \( \Phi(t) = B(t) \times 0.01 \pi \)
• Given: \( B(t) = B_0 e^{-t/\tau} \)

This gives the overall expression for magnetic flux over time in the loop.

Induced EMF

Faraday's law of electromagnetic induction is an essential principle for understanding how electrical currents can be generated from changing magnetic fields. It states that a change in magnetic flux through a circuit induces an electromotive force (EMF), which is essentially the voltage generated.

In practical terms, imagine waving your hand through a stream; your movement generates ripples, similar to how changing magnetic flux induces an EMF. This induced EMF can be calculated using the formula: \( \mathcal{E} = -N \frac{d\Phi}{dt} \), where \( N \) is the number of turns in the loop. The negative sign indicates the direction of the induced EMF opposes the change in flux.

• Formula: \( \mathcal{E} = 18 \times (-B_0 e^{-t/\tau} \frac{1}{\tau} \times 0.01\pi) \)
• Derivative: \( \frac{d\Phi}{dt} = -B_0 e^{-t/\tau} \frac{1}{\tau} \times 0.01\pi \)

The EMF is a crucial part of converting a changing magnetic environment into usable electrical energy.

Power Dissipation

Power dissipation in electrical circuits refers to the rate at which energy is used or lost in a system, usually as heat. Think of it as the warmth you feel from a lightbulb; that's power dissipating as heat.

To find the power dissipated over time, we use \( P(t) = I^2(t) R \), where \( I(t) \) is the current in the circuit and \( R \) is the resistance. Current itself is derived using Ohm's Law from the induced EMF we calculated.

• Power Formula: \( P(t) = \left(\frac{18 \times (-B_0 e^{-t/\tau} \frac{1}{\tau} \times 0.01\pi)}{2}\right)^2 \times 2 \)
• Effects: Power dissipated manifests as heat or light, impacting how devices perform.

Analyzing power dissipation helps understand energy conversion efficiency in systems involving electromagnetism.

Ohm's Law

Ohm's Law is a fundamental aspect of electrical engineering and physics, relating the three important electrical quantities: voltage (V), current (I), and resistance (R). Simplified as \( V = IR \), it states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance.

In our exercise, we use Ohm’s Law to calculate the current \( I(t) \) driven by the induced EMF through the loop of wire, which has a known resistance of \( 2 \Omega \).

• Formula: \( I(t) = \frac{\mathcal{E}}{R} \)
• Application: Determines the current flow based on calculated EMF and given resistance.

Understanding Ohm's Law is crucial for analyzing and designing electrical circuits, ensuring safe and effective power usage.