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Chapter 29: Problem 6

CALC A coil 4.00 \(\mathrm{cm}\) in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B=(0.0120 \mathrm{T} / \mathrm{s}) t+\left(3.00 \times 10^{-5} \mathrm{T} / \mathrm{s}^{4}\right) t^{4} .\) The coil is connected to a \(600-\Omega\) resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time \(t=5.00 \mathrm{s?}\)

Short Answer

Expert verified
The induced emf is \(\mathcal{E}(t) = -0.030156 - 3.0156 \times 10^{-4} t^3\), and the current at \(t = 5\) s is approximately \(-5.654 \times 10^{-5}\) A.

Step by step solution

01

Determine the Area and Turns of the Coil

First, calculate the area \(A\) of the coil. The radius \(r\) is given as 4.00 cm, which needs to be converted to meters: \(r = 0.04\) m. The area \(A\) of a circle is given by the formula \(A = \pi r^2\). Substituting the radius, we have \(A = \pi (0.04)^2 = 5.0265 \times 10^{-3} \text{ m}^2\). The number of turns \(N\) is 500.
02

Write the Expression for Magnetic Flux

The magnetic flux \(\Phi\) through the coil is given by \(\Phi = N \times A \times B(t)\), where \(B(t)\) is the given magnetic field. Therefore, the expression for \(\Phi\) becomes \(\Phi = 500 \times 5.0265 \times 10^{-3} \times \left((0.0120 \text{ T/s})t + (3.00 \times 10^{-5} \text{ T/s}^4)t^4\right)\).
03

Derive the Induced EMF

Using Faraday's law of electromagnetic induction, the induced emf \(\mathcal{E}\) is given by the rate of change of magnetic flux: \(\mathcal{E} = -\frac{d\Phi}{dt}\). Substitute \(\Phi\) from the previous step: \(\mathcal{E} = -500 \times 5.0265 \times 10^{-3} \times \frac{d}{dt}\big((0.0120 \text{ T/s})t + (3.00 \times 10^{-5} \text{ T/s}^4)t^4\big)\).
04

Compute the Time Derivative

Calculate the derivative: \(\frac{d}{dt}\big((0.0120 \text{ T/s})t + (3.00 \times 10^{-5} \text{ T/s}^4)t^4\big) = 0.0120 + 4 \times (3.00 \times 10^{-5})t^3\). Substituting, we have \(\mathcal{E} = -500 \times 5.0265 \times 10^{-3} \times (0.0120 + 4 \times (3.00 \times 10^{-5})t^3)\).
05

Simplify the Expression for Induced EMF

Multiply out the constants: \(\mathcal{E} = -2.513 \times (0.0120 + 1.2 \times 10^{-4} t^3)\). Simplifying further, \(\mathcal{E} = -2.513 \times 0.0120 - 2.513 \times 1.2 \times 10^{-4} t^3\), so \(\mathcal{E}(t) = -0.030156 - 3.0156 \times 10^{-4} t^3\).
06

Calculate Current at t = 5.00 s

Using Ohm's Law, \(I = \frac{\mathcal{E}}{R}\), where \(R = 600 \Omega\). Calculate \(\mathcal{E}(5) = -0.030156 - 3.0156 \times 10^{-4} \times (5.00)^3 = -0.030156 - 0.00376875\). Therefore, \(\mathcal{E}(5) = -0.03392475 \text{ V}\).Substituting into Ohm鈥檚 Law: \(I = \frac{-0.03392475}{600} = -5.654 \times 10^{-5} \text{ A}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Flux
Magnetic flux is a key concept in understanding electromagnetic induction. It represents the quantity of magnetic field passing through a given area and is denoted by \( \Phi \). In simpler terms, it measures how much of the magnetic field penetrates a particular surface.
To calculate the magnetic flux through a coil, one uses the formula \( \Phi = N \times A \times B(t) \), where:
  • \(N\) is the number of turns in the coil,
  • \(A\) is the area of the coil, and
  • \(B(t)\) is the magnetic field that varies with time.
Understanding how to compute magnetic flux is crucial, as it plays a pivotal role in determining the induced electromotive force (EMF).
Electromotive Force (EMF)
Electromotive Force, commonly referred to as EMF, is the energy provided per charge by an energy source in a circuit. Faraday's Law of Induction gives us the method to determine the induced EMF in a system, which is critical when the magnetic environment changes.
The induced EMF \( \mathcal{E} \) is calculated by the negative rate of change of magnetic flux, or mathematically, \( \mathcal{E} = -\frac{d\Phi}{dt} \). Here:
  • The negative sign is due to Lenz's Law, indicating that induced EMF will oppose the change in flux.
  • \( \frac{d\Phi}{dt} \) is the derivative of magnetic flux over time.
This concept is essential in creating and understanding current and voltage in circuits influenced by changing magnetic fields.
Ohm's Law
Ohm's Law is a fundamental principle used to relate voltage, current, and resistance in an electrical circuit. The law is typically expressed as \( V = I \times R \), where:
  • \(V\) is the voltage (or EMF in some contexts),
  • \(I\) is the current, and
  • \(R\) is the resistance.
In our problem, after calculating the induced EMF, Ohm鈥檚 Law helps us find the current flowing through the resistor. The current \( I \) can be directly calculated using \( I = \frac{\mathcal{E}}{R} \).
Understanding how Ohm鈥檚 Law connects these electrical quantities is crucial for effectively analyzing electric circuits.
Time-varying Magnetic Field
Time-varying magnetic fields are magnetic fields that change with time, which is central to the concepts of electromagnetic induction. In the exercise, the magnetic field is given as \( B(t) = (0.0120 \, \text{T/s})t + (3.00 \times 10^{-5} \, \text{T/s}^4)t^4 \).
  • The coefficient of \(t\) indicates how the magnetic field changes linearly over time.
  • The term with \(t^4\) suggests a higher-order time dependency, meaning more complex variation over time.
Such time-dependent behaviors require us to carefully differentiate and analyze their impact on magnetic flux and induced EMF.
Resistor in an Electric Circuit
A resistor in an electric circuit is an essential component used to limit the flow of electric current. It does this by providing resistance, which is measured in ohms (\( \Omega \)).
In this exercise, the resistor value \( R \) is \( 600 \Omega \). The current through the resistor, influenced by the induced EMF, is calculated using Ohm鈥檚 Law, \( I = \frac{\mathcal{E}}{R} \).
  • The presence of a resistor determines how much current can flow for a given voltage.
  • It is a fundamental part of controlling and managing electrical circuits efficiently.

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Most popular questions from this chapter

A 25.0 -cm-long metal rod lies in the \(x y\) -plane and makes an angle of \(36.9^{\circ}\) with the positive \(x\) -axis and an angle of \(53.1^{\circ}\) with the positive \(y\) -axis. The rod is moving in the \(+x\) -direction with a speed of 6.80 \(\mathrm{m} / \mathrm{s}\) . The rod is in a uniform magnetic field \(\vec{B}=(0.120 \mathrm{T}) \hat{i}-(0.220 \mathrm{T}) \hat{J}-(0.0900 \mathrm{T}) \hat{k}\) (a) What is the magnitude of the emf induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential.

CALC A conducting rod with length \(L=0.200 \mathrm{m},\) mass \(m=0.120 \mathrm{kg},\) and resistance \(R=80.0 \Omega\) moves without friction on metal rails as shown in Fig. \(29.11 .\) A uniform magnetic field with magnitude \(B=1.50 \mathrm{T}\) is directed into the plane of the figure. The rod is initially at rest, and then a constant force with magnitude \(F=1.90 \mathrm{N}\) and directed to the right is applied to the bar. How many seconds after the force is applied does the bar reach a speed of \(25,0 \mathrm{m} / \mathrm{s} ?\)

Cp Antenna emf. A satellite, orbiting the earth at the equator at an altitude of \(400 \mathrm{km},\) has an antenna that can be modeled as a 2.0 -m-long rod. The antenna is oriented perpendicular to the earth's surface. At the equator, the earth's magnetic field is essentially horizontal and has a value of \(8.0 \times 10^{-5} \mathrm{T}\) ; ignore any changes in \(B\) with altitude. Assuming the orbit is circular, determine the induced emf between the tips of the antenna.

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 \(\mathrm{cm} .\) The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 \(\mathrm{mV}\) ?

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 \(\mathrm{cm}^{2}\) is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is \(6.0 \times 10^{-5} \mathrm{T}\) . (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?
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