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Chapter 22: Problem 52

During a 72 -ms interval, a change in the current in a primary coil occurs. This change leads to the appearance of a \(6.0-\mathrm{mA}\) current in a nearby secondary coil. The secondary coil is part of a circuit in which the resistance is \(12 \Omega\). The mutual inductance between the two coils is \(3.2 \mathrm{mH}\). What is the change in the primary current?

Short Answer

Expert verified
The change in the primary current is 1.62 A.

Step by step solution

01

Understanding Mutual Inductance

Mutual inductance (M) is the property where a change in current in one coil induces a voltage in another coil through electromagnetic induction. The formula that relates mutual inductance, the induced current (I_s), and resistance (R) in the secondary coil is given by:\[ V_s = -M \left( \frac{dI_p}{dt} \right) \]Where V_s is the induced voltage in the secondary coil and \frac{dI_p}{dt} is the rate of change of current in the primary coil.
02

Calculate Induced Voltage

The secondary coil carries a current of 6.0 \, \mathrm{mA} (or 0.006 \, \mathrm{A}) with a resistance of 12 \, \Omega, thus the induced voltage (V_s) in the secondary coil can be calculated using Ohm's Law:\[ V_s = I_s \times R \]Substitute the given values:\[ V_s = 0.006 \, \mathrm{A} \times 12 \, \Omega = 0.072 \, \mathrm{V} \]
03

Relate Induced Voltage to Rate of Change of Current

Utilize the relation between induced voltage and the rate of change of current from Step 1:\[ V_s = -M \left( \frac{dI_p}{dt} \right) \]Substitute V_s = 0.072 \, \mathrm{V} and M = 3.2 \, \mathrm{mH} = 0.0032 \, \mathrm{H}:\[ 0.072 = 0.0032 \times \left( \frac{dI_p}{dt} \right) \]
04

Solve for the Change in Primary Current

Rearrange the equation from Step 3 to solve for the rate of change of primary current:\[ \frac{dI_p}{dt} = \frac{0.072}{0.0032} \]Calculate the result:\[ \frac{dI_p}{dt} = 22.5 \, \mathrm{A/s} \]
05

Calculate Actual Change in Current

The rate of change of current \left( \frac{dI_p}{dt} \right) over a time interval \Delta t = 72 \, \mathrm{ms} = 0.072 \, \mathrm{s} gives us the total change in current (\Delta I_p):\[ \Delta I_p = \frac{dI_p}{dt} \times \Delta t \]Substitute the values:\[ \Delta I_p = 22.5 \, \mathrm{A/s} \times 0.072 \, \mathrm{s} = 1.62 \, \mathrm{A} \]

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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 phenomenon where a change in magnetic field, usually due to a varying electric current, generates an electromotive force (EMF) in a nearby conductor. This foundational concept in physics was discovered by Michael Faraday in 1831. It explains how electric currents can be induced in a coil by changing currents in another coil placed nearby. This is primarily what happens in transformers and many other electrical devices.

One key element is **Faraday's Law of Induction**, which states that the induced EMF in a coil is proportional to the rate of change of the magnetic flux through the coil. This is mathematically represented as \( \varepsilon = -N \frac{d\Phi}{dt} \), where \( \varepsilon \) is the induced EMF, \( N \) is the number of loops, and \( \frac{d\Phi}{dt} \) is the rate of change of the magnetic flux. The negative sign represents Lenz's Law, indicating that the induced EMF will oppose the change in the magnetic flux.

Electromagnetic induction is essential for understanding mutual inductance, which is the process used in the exercise to determine how the change in current in a primary coil affects a secondary coil.
Ohm's Law
Ohm's Law is a fundamental principle used to relate the current flowing through a conductor with the voltage across it and its resistance. Simply put, it states that the current through a conductor between two points is directly proportional to the voltage across the two points. The mathematical representation of Ohm's Law is \( V = I \times R \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance.

This principle was used in the original exercise to calculate the induced voltage in the secondary coil. Given the current and resistance in the secondary coil, Ohm's Law allowed the calculation of this induced voltage, providing crucial data for solving the mutual inductance problem.

By understanding Ohm's Law, students can connect how electric circuits behave and predict how voltage, current, and resistance interact in various parts of a circuit.
Primary Coil
In the context of transformers and mutual inductance, the primary coil is the coil that initially receives an electrical input. With a change in current, it generates a magnetic field affecting the secondary coil through electromagnetic induction.

The role of the primary coil in mutual inductance is to be the source of the changing magnetic field, which is why it's essential in calculating the induced EMF and, subsequently, the current in the secondary coil. The primary coil's current change impacts the magnetic field and, through mutual inductance, affects the secondary coil's electrical characteristics.

In practical terms, understanding how a primary coil operates is fundamental when designing circuits involving transformers or inductors. It helps illustrate the impact of one circuit on another, balancing energy transfer without direct electrical connection.

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Most popular questions from this chapter

A piece of copper wire is formed into a single circular loop of radius \(12 \mathrm{cm} .\) A magnetic field is oriented parallel to the normal to the loop. and it increases from 0 to \(0.60 \mathrm{T}\) in a time of \(0.45 \mathrm{s}\). The wire has a resistance per unit length of \(3.3 \times 10^{-2} \Omega / \mathrm{m} .\) What is the average electrical energy dissipated in the resistance of the wire?

A \(0.80-\mathrm{m}\) aluminum bar is held with its length parallel to the east- west direction and dropped from a bridge. Just before the bar hits the river below, its speed is \(22 \mathrm{m} / \mathrm{s},\) and the emf induced across its length is \(6.5 \times 10^{-4} \mathrm{V} .\) Assuming the horizontal component of the earth's magnetic field at the location of the bar points directly north, (a) determine the magnitude of the horizontal component of the earth's magnetic field, and (b) state whether the east end or the west end of the bar is positive.

Consult Multiple-Concept Example 11 for background material relating to this problem. A small rubber wheel on the shaft of a bicycle generator presses against the bike tire and turns the coil of the generator at an angular speed that is 38 times as great as the angular speed of the tire itself. Each tire has a radius of \(0.300 \mathrm{m}\). The coil consists of 125 turns, has an area of \(3.86 \times 10^{-3} \mathrm{m}^{2},\) and rotates in a \(0.0900-\mathrm{T}\) magnetic field. The bicycle starts from rest and has an acceleration of \(+0.550 \mathrm{m} / \mathrm{s}^{2} .\) What is the peak emf produced by the generator at the end of 5.10 s?

A \(120.0-\mathrm{V}\) motor draws a current of \(7.00 \mathrm{A}\) when running at normal speed. The resistance of the armature wire is \(0.720 \Omega .\) (a) Determine the back emf generated by the motor. (b) What is the current at the instant when the motor is just turned on and has not begun to rotate? (c) What series resistance must be added to limit the starting current to \(15.0 \mathrm{A} ?\)

A magnetic field is passing through a loop of wire whose area is \(0.018 \mathrm{m}^{2}\). The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of \(0.20 \mathrm{T} / \mathrm{s}\) (a) Determine the magnitude of the emf induced in the loop. (b) Suppose that the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in \(\mathrm{m}^{2} / \mathrm{s}\) ) should the area be changed at the instant when \(B=1.8 \mathrm{T}\) if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.
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