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Chapter 22: Problem 49

During a 72 -ms interval, a change in the current in a primary coil occurs. This change leads to the appearance of a \(6.0\) -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 \(-1.62\, \text{A}\).

Step by step solution

01

Understand the problem concept

We are dealing with mutual inductance, where a change in current in a primary coil induces a current in a nearby secondary coil. We need to find the change in the primary current using the given data.
02

Identify the formula

The formula that relates mutual inductance, change in current, and induced electromotive force (EMF) is given by:\[ \varepsilon = -M \frac{\Delta I_p}{\Delta t} \]where \( M \) is the mutual inductance, \( \Delta I_p \) is the change in current in the primary coil, and \( \Delta t \) is the time interval.
03

Calculate the induced EMF in the secondary coil

The induced EMF (\( \varepsilon \)) in the secondary coil is calculated using Ohm's Law, \( \varepsilon = IR \), where \( I \) is the current and \( R \) is the resistance.Given \( I = 6.0 \, \text{mA} = 0.006 \, \text{A} \) and \( R = 12 \Omega \),\[ \varepsilon = 0.006 \times 12 = 0.072 \text{ V} \]
04

Rearrange the formula to find \( \Delta I_p \)

We need to find \( \Delta I_p \), so rearrange the formula:\[ \Delta I_p = -\frac{\varepsilon \cdot \Delta t}{M} \]
05

Substitute and solve

Substitute the values of \( \varepsilon = 0.072 \text{ V} \), \( \Delta t = 72 \text{ ms} = 0.072 \text{ s} \), and \( M = 3.2 \text{ mH} = 3.2 \times 10^{-3} \text{ H} \) into the equation:\[ \Delta I_p = -\frac{0.072 \times 0.072}{3.2 \times 10^{-3}} \]\[ \Delta I_p = -1.62 \, \text{A} \]
06

Interpret the result

The negative sign indicates the direction of the change, meaning the direction of the primary current change is opposite to the assumed direction.

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Key Concepts

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

Primary Coil
The primary coil is a key component in a system involving mutual inductance. It is the coil where changes in the electrical current originate. These changes are crucial because they lead to electromagnetic effects in nearby circuits. When the current in the primary coil changes, it creates a magnetic field that can influence another coil. This is a fundamental concept in transformers, where the primary coil is responsible for inducing a voltage in a secondary coil. By understanding how the primary coil operates, we can predict the effects it will have on nearby coils.
Secondary Coil
A secondary coil is positioned close to the primary coil and is affected by changes in the magnetic field produced by the primary coil. This coil experiences an induced current due to a phenomenon known as electromagnetic induction. The current induced in the secondary coil is what allows energy transfer from the primary coil to occur. This process is essential in many electrical devices, as it enables the secondary coil to produce an electrical current without direct contact with the source of the magnetic field. Therefore, the design and placement of the secondary coil are crucial for efficient energy transfer and for maximizing the system's effectiveness.
Induced Electromotive Force
Induced electromotive force (EMF) is the voltage generated in a circuit due to a change in magnetic field. In the context of mutual inductance, this change is a result of the alternating current in the primary coil. The induced EMF is crucial because it drives the current in the secondary coil. You can calculate it using Ohm's Law, where the EMF is the product of the current and resistance in the circuit. The concept of induced EMF is a central element that explains how energy is transferred between coils in a transformer without a physical connection, just through magnetic fields.
Ohm's Law
Ohm's Law is a fundamental principle in electronics that relates voltage (V), current (I), and resistance (R) in a circuit. The law is often written as \( V = IR \), meaning the voltage across the circuit is equal to the product of the current and the resistance. In the context of mutual inductance, Ohm's Law is used to determine the induced EMF in the secondary coil. By knowing the current flowing through the secondary coil and its resistance, you can easily compute the induced voltage. Ohm's Law helps convert measurable quantities into useful information for analyzing and designing various electronic circuits, making it indispensable in understanding how changes in one part of a circuit affect the rest.

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Most popular questions from this chapter

A circular coil \((950\) turns, radius \(=0.060 \mathrm{~m})\) is rotating in a uniform magnetic field. At \(t=0 \mathrm{~s}\), the normal to the coil is per pendicular to the magnetic field. At \(t=0.010 \mathrm{~s}\) the normal makes an angle of \(\phi=45^{\circ}\) with the field because the coil has made oneeighth of a revolution. An average emf of magnitude \(0.065 \mathrm{~V}\) is induced in the coil. Find the magnitude of the magnetic field at the location of the 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.

A planar coil of wire has a single turn. The normal to this coil is parallel to a uniform and constant (in time) magnetic field of \(1.7 \mathrm{~T}\). An emf that has a magnitude of \(2.6 \mathrm{~V}\) is induced in this coil because the coil's area \(A\) is shrinking. What is the magnitude of \(\Delta A / \Delta t,\) which is the rate \(\left(\right.\) in \(\left.m^{2} / s\right)\) at which the area changes?

Concept Questions The drawing shows a coil of copper wire that consists of two semicircles joined by straight sections of wire. In part \(a\) the coil is lying flat on a horizontal surface. The dashed line also lies in the plane of the horizontal surface. Starting from the orientation in part \(a,\) the smaller semicircle rotates at an angular frequency \(\omega\) about the dashed line, until its plane becomes perpendicular to the horizontal surface, as shown in part \(b\). A uniform magnetic field is constant in time and is directed upward, perpendicular to the horizontal surface. The field completely fills the region occupied by the coil in either part of the drawing. (a) In which part of the drawing, if either, does a greater magnetic flux pass through the coil? Account for your answer. (b) As the shape of the coil changes from that in part \(a\) of the drawing to that in part \(b\), does an induced current flow in the coil, and, if so, in which direction does it flow? Give your reasoning. To describe the flow, imagine that you are above the coil looking down at it. (c) How is the period \(T\) of the rotational motion related to the angular frequency \(\omega\), and in terms of the period, what is the shortest time interval that elapses between parts \(a\) and \(b\) of the drawing? Problem The magnitude of the magnetic field is \(0.35 \mathrm{~T}\). The resistance of the coil is \(0.025 \Omega,\) and the smaller semicircle has a radius of \(0.20 \mathrm{~m} .\) The angular frequency at which the small semicircle rotates is \(1.5 \mathrm{rad} / \mathrm{s} .\) Determine the average current, if any, induced in the coil as the coil changes shape from that in part \(a\) of the drawing to that in \(\operatorname{part} b\)

A generating station is producing \(1.2 \times 10^{6} \mathrm{~W}\) of power that is to be sent to a small town located \(7.0 \mathrm{~km}\) away. Each of the two wires that comprise the transmission line has a resistance per kilometer of length of \(5.0 \times 10^{-2} \Omega / \mathrm{km} .\) (a) Find the power used to heat the wires if the power is transmitted at \(1200 \mathrm{~V} .\) (b) A \(100: 1\) step-up transformer is used to raise the voltage before the power is transmitted. How much power is now used to heat the wires?
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