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Chapter 21: Problem 29

Two coils are wound around the same cylindrical form, like the coils in Example \(21.8 .\) When the current in the first coil is decreasing at a rate of \(0.242 \mathrm{A} / \mathrm{s},\) the induced emf in the second coil has magnitude 1.65 \(\mathrm{mV}\) . (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the average magnetic flux through each turn when the current in the first coil equals 1.20 \(\mathrm{A} ?(\mathrm{c})\) If the current in the second coil increases at a rate of \(0.360 \mathrm{A} / \mathrm{s},\) what is the magnitude of the induced emf in the first coil?

Short Answer

Expert verified
(a) \( M \approx 6.82 \times 10^{-3} \,\mathrm{H} \); (b) \( \Phi_{21} \approx 3.27 \times 10^{-4} \,\mathrm{Wb} \); (c) \( \mathcal{E}_{12} \approx 2.46 \,\mathrm{mV} \).

Step by step solution

01

Understanding the Given Data for Part (a)

We are given that the rate of change of current in the first coil \( \frac{dI_1}{dt} = -0.242 \,\mathrm{A/s} \) and the induced emf in the second coil \( \mathcal{E}_{21} = 1.65 \,\mathrm{mV} = 1.65 \times 10^{-3} \,\mathrm{V} \). Using the mutual inductance formula \( \mathcal{E}_{21} = M \frac{dI_1}{dt} \), we can solve for \( M \).
02

Calculating the Mutual Inductance for Part (a)

Rearrange the formula to solve for \( M \):\[ M = \frac{\mathcal{E}_{21}}{\frac{dI_1}{dt}} = \frac{1.65 \times 10^{-3}}{-0.242} \,\mathrm{H}. \]Calculate \( M \) using these values.
03

Calculation Result for Mutual Inductance

Substituting the values, we get:\[ M = \frac{1.65 \times 10^{-3}}{0.242} \approx 6.82 \times 10^{-3} \,\mathrm{H}. \] This is the mutual inductance of the pair of coils.
04

Understanding the Given Data for Part (b)

We know \( N_2 = 25 \), the number of turns in the second coil. We need to find the average magnetic flux \( \Phi_{21} \) through each turn when the current in the first coil is \( 1.20 \,\mathrm{A} \). Use the formula \( \Phi_{21} = \frac{M I_1}{N_2} \) where \( I_1 = 1.20 \,\mathrm{A} \).
05

Calculating the Magnetic Flux for Part (b)

Using the mutual inductance \( M = 6.82 \times 10^{-3} \,\mathrm{H} \), we find:\[ \Phi_{21} = \frac{6.82 \times 10^{-3} \times 1.20}{25}. \] Calculate \( \Phi_{21} \) using this expression.
06

Calculation Result for Magnetic Flux

Substituting the values, we get:\[ \Phi_{21} = \frac{6.82 \times 10^{-3} \times 1.20}{25} \approx 3.27 \times 10^{-4} \,\mathrm{Wb} \] per turn.
07

Understanding the Given Data for Part (c)

The rate of change of current in the second coil is \( \frac{dI_2}{dt} = 0.360 \,\mathrm{A/s} \). We need to find the induced emf in the first coil \( \mathcal{E}_{12} \) using \( \mathcal{E}_{12} = M \frac{dI_2}{dt} \).
08

Calculating the Induced EMF for Part (c)

Using the mutual inductance \( M = 6.82 \times 10^{-3} \,\mathrm{H} \), we find:\[ \mathcal{E}_{12} = 6.82 \times 10^{-3} \times 0.360. \]Calculate \( \mathcal{E}_{12} \) using these values.
09

Calculation Result for Induced EMF

Substituting the values, we get:\[ \mathcal{E}_{12} = 6.82 \times 10^{-3} \times 0.360 \approx 2.46 \times 10^{-3} \,\mathrm{V} = 2.46 \,\mathrm{mV}. \]This is the magnitude of the induced emf in the first coil.

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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 measure of the amount of magnetic field passing through an area. It is an essential concept in understanding how magnetic fields interact with materials, like coils in this exercise. Magnetic flux (\( \Phi \) ) is calculated by multiplying the magnetic field (\( B \) ) by the perpendicular area (\( A \) ) through which the field passes:
  • \[\Phi = B \cdot A \\]
  • The unit of magnetic flux is Webers (Wb).
In the context of coils, as given in the exercise, the magnetic flux through each turn of the coil is concerned. Here, it's calculated using mutual inductance (\( M \) ) and the current in the coil:
  • \[\Phi = \frac{MI}{N} \\]
  • where \( I \) is the current and \( N \) is the number of turns in the coil.
Understanding magnetic flux helps in determining how efficiently a coil can induce an electromotive force when exposed to changing magnetic fields.
Faraday's Law of Induction
Faraday's Law of Induction explains how a changing magnetic flux can induce an electromotive force (emf) in a circuit. It is one of the foundational principles of electromagnetism. Essentially, it describes how the emf generated is related to the rate of change of magnetic flux through a circuit:
  • \[ \mathcal{E} = - \frac{d\Phi}{dt} \]
  • The negative sign signifies Lenz's law, indicating the direction of the induced emf opposes the change in flux.
  • This principle is critical in understanding transformers, electric generators, and inductors.
In the exercise, Faraday's Law helps calculate the induced emf in a secondary coil when there is a change in current in a primary coil. This relationship is quantifiable due to the mutual inductance between the coils.
Induced EMF
An induced electromotive force (emf) is generated when there is a change in magnetic environment around a conductor, like the coils in our exercise. According to Faraday's Law, this induced emf (\( \mathcal{E} \) ) is directly dependent on the rate at which the magnetic flux changes. For two mutually coupled coils, the expression becomes:
  • \[ \mathcal{E} = M \frac{dI}{dt} \]
  • where \( M \) is the mutual inductance and \( \frac{dI}{dt} \) is the rate of change of current.
This concept is precisely how transformers work, allowing power transmission over long distances. In the given problem, it explains how a changing current in one coil results in an emf in the other coil, emphasizing the importance of mutual inductance. Understanding this helps grasp how energy transfer occurs in electromagnetic fields.

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Most popular questions from this chapter

A circular area with a radius of 6.50 \(\mathrm{cm}\) lies in the \(x\) -y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field \(B=0.230 \mathrm{T}\) that points (a) in the \(+z\) direction? (b) at an angle of \(53.1^{\circ}\) from the \(+z\) direction? (c) in the \(+y\) direction?

A flat, square coil with 15 turns has sides of length 0.120 \(\mathrm{m}\) . The coil rotates in a magnetic field of 0.0250 \(\mathrm{T}\) (a) What is the angular velocity of the coil if the maximum emf produced is 20.0 \(\mathrm{mV}^{\prime} ?\) (Hint: Look at the motional emf induced across the ends of the segments of the coil.) (b) What is the average emf at this angular velocity?

When a certain inductor carries a current \(I,\) it stores 3.0 \(\mathrm{mJ}\) of magnetic energy. How much current (in terms of \(I )\) would it have to carry to store 9.0 \(\mathrm{mJ}\) of energy?

You're driving at 95 \(\mathrm{km} / \mathrm{h}\) in a direction \(35^{\circ}\) east of north, in a region where the earth's magnetic field of \(5.5 \times 10^{-5} \mathrm{T}\) is horizontal and points due north. If your car measures 1.5 \(\mathrm{m}\) from its underbody to its roof, calculate the induced emf between roof and underbody. (You can assume the sides of the car are straight and vertical.) Is the roof of the car at a higher or lower potential than the underbody?

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 \(\mathrm{cm}^{2}\) is rotated from a position where its plane is perpendicular to the earth's magnetic field to one where its plane is parallel to the field. The rotation takes 0.040 s. The earth's magnetic field at the location of the laboratory 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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