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Chapter 22: Problem 51

Two coils of wire are placed close together. Initially, a current of \(2.5 \mathrm{A}\) exists in one of the coils, but there is no current in the other. The current is then switched off in a time of \(3.7 \times 10^{-2} \mathrm{s}\). During this time, the average emf induced in the other coil is \(1.7 \mathrm{V}\). What is the mutual inductance of the two-coil system?

Short Answer

Expert verified
The mutual inductance of the two-coil system is approximately 0.0252 H.

Step by step solution

01

Identify Given Values

We have the initial current in the first coil \( I = 2.5 \, \text{A} \), the time interval for the current to switch off \( \Delta t = 3.7 \times 10^{-2} \text{s} \), and the average induced emf in the second coil \( \varepsilon = 1.7 \, \text{V} \).
02

Understand the Formula for Mutual Inductance

Mutual inductance \( M \) is related to the change in current and the induced emf by the formula: \( \varepsilon = -M \frac{\Delta I}{\Delta t} \). Rearranging to solve for \( M \), we get \( M = -\frac{\varepsilon \cdot \Delta t}{\Delta I} \).
03

Calculate Change in Current

The change in current \( \Delta I \) is equal to the initial current, since the current goes from 2.5 A to 0 A. Thus, \( \Delta I = 2.5 \, \text{A} - 0 \, \text{A} = 2.5 \, \text{A} \).
04

Substitute Values into the Formula

Substitute the known values into the rearranged mutual inductance formula: \( M = -\frac{1.7 \, \text{V} \cdot 3.7 \times 10^{-2} \, \text{s}}{2.5 \, \text{A}} \).
05

Perform the Calculation

Calculate the result: \( M = -\frac{1.7 \, \times 3.7 \times 10^{-2}}{2.5} = -\frac{0.0629}{2.5} = -0.02516 \, \text{H} \).
06

Interpret the Result

The mutual inductance \( M \) is a magnitude, so its value is \( M = 0.02516 \, \text{H} \). The negative sign indicates the direction of the induced emf is opposite to the change in current, which is consistent with Lenz's law.

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Key Concepts

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

Electromagnetic Induction
Electromagnetic induction is a fascinating phenomenon where an electric current is produced in a conductor due to a changing magnetic field. This concept was discovered by Michael Faraday in the early 19th century. In simpler terms, if you have a coil of wire and you change the magnetic environment around it, or move it within a magnetic field, it can create a voltage across the coil.
Here are some key points:
  • Electromagnetic induction is the principle behind many electrical devices, such as transformers and electric generators.
  • It is a crucial concept in physics, as it explains how electricity can be generated from magnetic fields.
  • The strength of the induced current depends on the rate of change of the magnetic field and the number of turns in the coil.
Whenever you turn a magnet near a coil or move the coil in a magnetic field, you are utilizing electromagnetic induction to produce electricity. This is essentially how power generators work, converting mechanical energy into electrical energy through motion and magnetism.
Lenz's Law
Lenz's Law gives direction to the induced current or emf. It was formulated by Heinrich Lenz in 1834, providing an essential insight into the nature of electromagnetic induction. The law states that the direction of an induced electromagnetic force (emf) will be such that it opposes the change in magnetic flux that produced it.
Here's why Lenz's Law is so important:
  • This law ensures the conservation of energy. The induced current always works to cancel out the change that created it, preventing any free energy from being created.
  • By observing the direction of the induced emf, one can determine the direction of the changing magnetic field.
  • Lenz's Law is also consistent with Newton's Third Law of Motion 鈥� every action has an equal and opposite reaction.
In practice, if you push a magnet into a coil, the induced current will generate its own magnetic field that repels the magnet. If you pull the magnet away, the induced field will attract it back. This shows clearly how Lenz's Law works to maintain balance.
Induced EMF
Induced electromotive force (emf) is the voltage generated across a coil or circuit when it undergoes a change in magnetic environment. This is the effect predicted by electromagnetic induction. When a circuit experiences changes in the magnetic field around it, an emf is created.
Key aspects of induced emf include:
  • The faster the magnetic field changes, the larger the induced emf 鈥� this is a principle used in various technologies today.
  • Calculating induced emf is often done using Faraday's Law, which involves the number of turns in the coil and the rate of change of the magnetic flux.
  • The unit of emf is the volt (V), and induced emf can exist even without a complete loop, as it is sufficient that the magnetic environment changes.
In the mutual inductance problem, the induced emf in the second coil was generated by turning off the current quickly in the first coil, causing a swift change in the magnetic field experienced by the second coil. This demonstrates the practical application of these principles, showing the connection between physical changes and electrical responses.

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Most popular questions from this chapter

Indicate the direction of the electric field between the plates of the parallel plate capacitor shown in the drawing if the magnetic field is decreasing in time. Give your reasoning.

The coil of a generator has a radius of \(0.14 \mathrm{m} .\) When this coil is unwound, the wire from which it is made has a length of \(5.7 \mathrm{m}\). The magnetic field of the generator is \(0.20 \mathrm{T}\), and the coil rotates at an angular speed of \(25 \mathrm{rad} / \mathrm{s}\). What is the peak emf of this generator?

Coil 1 is a flat circular coil that has \(N_{1}\) turns and a radius \(R_{1}\). At its center is a much smaller flat, circular coil that has \(N_{2}\) turns and radius \(R_{2}\). The planes of the coils are parallel. Assume that coil 2 is so small that the magnetic field due to coil 1 has nearly the same value at all points covered by the area of coil \(2 .\) Determine an expression for the mutual inductance between these two coils in terms of \(\mu_{0}, N_{1}, R_{1}, N_{2},\) and \(R_{2}\)

In some places, insect "zappers," with their blue lights, are a familiar sight on a summer's night. These devices use a high voltage to electrocute insects. One such device uses an ac voltage of \(4320 \mathrm{V},\) which is obtained from a standard \(120.0-\mathrm{V}\) outlet by means of a transformer. If the primary coil has 21 turns, how many turns are in the secondary coil?

Near San Francisco, where the vertically downward component of the earth's magnetic field is \(4.8 \times 10^{-5} \mathrm{T}, \mathrm{a}\) car is traveling forward at \(25 \mathrm{m} / \mathrm{s}\) The width of the car is \(2.0 \mathrm{m}\). (a) Find the emf induced between the two sides of the car. (b) Which side of the car is positive - the driver's side or the passenger's side?
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