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Chapter 22: Problem 46

Two coils of wire are placed close together. Initially, a current of 2.5 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 is approximately 0.0251 Henry.

Step by step solution

01

Identify Known Values

Let's start by identifying the known values in the problem:- Initial current in the first coil, \(I_0 = 2.5\, \mathrm{A}\).- Final current, \(I_f = 0\, \mathrm{A}\).- Time over which the current changes, \(\Delta t = 3.7 \times 10^{-2}\, \mathrm{s}\).- Induced emf in the second coil, \(\varepsilon = 1.7\, \mathrm{V}\).
02

Use Formula for Mutual Inductance

The average induced emf in the second coil can be related to the rate of change of current in the first coil through their mutual inductance:\[ \varepsilon = -M \frac{\Delta I}{\Delta t} \]where \(M\) is the mutual inductance and \(\Delta I = I_f - I_0\) is the change in current in the first coil.
03

Calculate Change in Current

The change in current \(\Delta I\) in the first coil is given by:\[ \Delta I = I_f - I_0 = 0 - 2.5 = -2.5\, \mathrm{A} \]This is the amount by which the current decreases.
04

Solve for Mutual Inductance

Rearrange the formula for emf to solve for the mutual inductance \(M\):\[ M = -\frac{\varepsilon}{\Delta I / \Delta t} \]Substitute the values we have:\[ M = -\frac{1.7}{-2.5 / 3.7 \times 10^{-2}} \]Calculate:- \(\Delta I / \Delta t = -2.5 / 3.7 \times 10^{-2} = -67.57\, \mathrm{A/s}\)- Then \(M = 1.7 / 67.57 \approx 0.0251\, \mathrm{H}\).
05

Calculate Mutual Inductance

The mutual inductance \(M\) for the coil system is calculated as follows:\[ M = \frac{1.7}{67.57} = 0.0251\, \mathrm{H} \]

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Key Concepts

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

Induced EMF
Electromotive force, or EMF, is a voltage generated by a change in magnetic environment. When you have two coils placed close together, a change in the electric current in one coil affects the other. This is due to mutual inductance.
This concept results in the induction of a voltage, or EMF, in the second coil when the current that flows through the first coil changes.
  • Induced EMF plays a crucial role in the functioning of transformers, where it helps in electrical energy transfer.
  • Its magnitude depends on how fast the current in one coil changes, affecting the magnetic field and thus inducing EMF in the nearby coil.
The basic working principle relies on Faraday鈥檚 Law of Electromagnetic Induction, where EMF produced is directly proportional to the rate of change of magnetic flux.
Rate of Change of Current
The rate at which current changes in a coil is fundamental to understanding mutual inductance. In simple terms, when you switch a coil鈥檚 current on or off, it doesn鈥檛 happen instantaneously; it takes some time for the current to ramp up or down.
The rate of change of current ( \Delta I/ \Delta t ) in the coil was determined in the solution: \[-2.5 \, \text{A} / 3.7 \times 10^{-2} \, \text{s}\]. This yields a rate of \-67.57 \, \text{A/s}.
  • The magnitude of this change impacts the induced EMF in a second coil.
  • Understanding this concept is essential for designing circuits that have inductors, particularly in applications that require precise control over current changes.
Knowing the rate of change helps in predicting the magnitude of the induced voltage or EMF in a nearby coil.
Coil System
A coil system typically involves two or more coils placed close to one another, which means they interact via their magnetic fields. This setup is essential in devices like transformers and induction coils.
The two coils being placed close to one another in our exercise allow their magnetic fields to interact when the current is varied.
  • This interaction can either be constructive or destructive, depending on the direction of the current flow and can influence the efficacy of the induced EMF.
  • In practical terms, the positioning of coils can affect their mutual inductance value, determining how much EMF is induced.
Understanding coil systems not only helps in grasping mutual inductance but also in various practical applications like wireless chargers and electric motors.
Change in Current
Change in current is a critical parameter affecting both self and mutual inductance. In this exercise, the change in current is calculated as the difference between the final and initial current, \[ \Delta I = I_f - I_0 = 0 - 2.5 \, \text{A} = -2.5 \, \text{A} \. \]
This value indicates how the current drops from its initial to its final value within the given time frame.
  • This change is the reason for the induced EMF in the nearby coil.
  • The negative sign signifies a decrease in current, which is pertinent when determining the direction of the induced EMF based on Lenz鈥檚 Law.
Understanding changes in current is essential for accurately predicting how a coil will behave in a changing magnetic environment.

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Most popular questions from this chapter

A vacuum cleaner is plugged into a \(120.0-\mathrm{V}\) socket and uses 3.0 A of current in normal operation when the back emf generated by the electric motor is \(72.0 \mathrm{~V}\). Find the coil resistance of the motor.

A step-down transformer (turns ratio \(=1: 8\) ) is used with an electric train to reduce the voltage from the wall receptacle to a value needed to operate the train. When the train is running, the current in the secondary coil is \(1.6 \mathrm{~A}\). What is the current in the primary coil?

A generator is connected across the primary coil \(\left(N_{\mathrm{p}}\right.\) turns) of a transformer, while a resistance \(R_{2}\) is connected across the secondary coil \(\left(N_{\mathrm{s}}\right.\) turns \() .\) This circuit is equivalent to a circuit in which a single resistance \(R_{1}\) is connected directly across the generator, without the transformer. Show that \(R_{1}=\left(N_{\mathrm{p}} / N_{\mathrm{s}}\right)^{2} R_{2},\) by starting with Ohm's law as applied to the secondary coil.

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?

Concept Questions The drawing shows a straight wire carrying a current \(I\). Above the wire is a rectangular loop that contains a resistor \(R\). (a) Does the magnetic field produced by the current \(I\) penetrate the loop and generate a magnetic flux? (b) When is there an induced current in the loop, if the current \(I\) is constant or if it is decreasing in time? (c) When there is an induced magnetic field produced by the loop, does it always have a direction that is opposite to the direction of the magnetic field produced by the current \(I\) ? Provide a reason for each answer. Problem If the current \(I\) is decreasing in time, what is the direction of the induced current through the resistor \(R\) - left to right or right to left? Give your reasoning.
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