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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-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

Understand the Formula

The formula for mutual inductance when linking currents in primary and secondary coils is given by \( V_s = M \frac{\Delta I_p}{\Delta t} \), where \( V_s \) is the induced voltage in the secondary coil, \( M \) is the mutual inductance, \( \Delta I_p \) is the change in primary current, and \( \Delta t \) is the time interval.
02

Determine the Induced Voltage

The induced voltage \( V_s \) across the secondary coil can be calculated using Ohm鈥檚 Law: \( V_s = I_s \times R_s \), where \( I_s = 6.0 \, \text{mA} = 0.006 \, \text{A} \) and \( R_s = 12 \, \Omega \). Thus, \( V_s = 0.006 \times 12 \) volts.
03

Calculate Induced Voltage

Solve for \( V_s \) using \( V_s = 0.006 \times 12 \). This results in \( V_s = 0.072 \text{ V} \).
04

Use the Mutual Inductance Formula

Rearrange \( V_s = M \frac{\Delta I_p}{\Delta t} \) to solve for the change in primary current \( \Delta I_p \): \( \Delta I_p = \frac{V_s \cdot \Delta t}{M} \). Plug in \( V_s = 0.072 \text{ V} \), \( \Delta t = 72 \text{ ms} = 0.072 \text{ s} \), and \( M = 3.2 \text{ mH} = 0.0032 \text{ H} \).
05

Compute the Change in Primary Current

Substitute the values into \( \Delta I_p = \frac{0.072 \times 0.072}{0.0032} \). Calculate to find \( \Delta I_p = 1.62 \text{ A} \).

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Key Concepts

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

Ohm's Law
Ohm's Law is a fundamental principle used in electricity that relates voltage, current, and resistance. It is expressed through the formula, \( V = I \times R \), where \( V \) stands for voltage, \( I \) for current, and \( R \) for resistance. This law shows how these three elements are interdependent.

In essence, Ohm's Law tells us that the voltage in a circuit is equal to the current it carries multiplied by its resistance. This concept is crucial for analyzing circuits, especially when determining key variables like current or voltage in different components.
  • Voltage (\( V \)) is the push that drives electric charge around a circuit.
  • Current (\( I \)) is the rate of flow of electric charge.
  • Resistance (\( R \)) is the opposition that a material offers to the flow of charge.
In our problem, Ohm's Law is used to calculate the induced voltage in the secondary coil circuit. By multiplying the current through the secondary coil (6.0 mA or 0.006 A) by its resistance (12 \( \Omega \)), we find the voltage to be \( V_s = 0.072 \) V.
Induced Voltage
Induced voltage, or electromotive force (EMF), is generated in a coil when there is a change in the magnetic field around it. This is a basic principle of electromagnetism known as electromagnetic induction.

The concept is vital in the operation of transformers, generators, and various other electrical devices. It typically depends on factors such as the number of turns in a coil and the rate of change of the magnetic field.
  • It is calculated using the formula \( V_s = I_s \times R_s \).
  • In our example, an induced voltage of 0.072 V is calculated in the secondary coil.
Here, the changing primary current causes an electromagnetic field, inducing voltage across the secondary coil, which we calculated using Ohm's Law. This induced voltage is crucial for transferring energy from the primary to the secondary coil.
Primary Current Change
The primary current change is a critical concept in understanding mutual inductance, particularly in how a change in current in one coil affects another in a nearby coil. When there is a change in the primary current, it changes the magnetic flux, thus inducing a voltage (EMF) across the secondary coil.

The change in primary current is derived with the formula for mutual inductance: \( V_s = M \cdot \frac{\Delta I_p}{\Delta t} \), rearranging this allows solving for \( \Delta I_p \).
  • This value is determined to be 1.62 A in the specific problem provided.
  • The mutual inductance given is 3.2 mH, allowing us to determine how changes in the primary affect the secondary circuit.
By knowing the induced voltage and the mutual inductance, we calculated how much the primary current had to change to result in the observed effects on the secondary coil. This relationship is foundational for understanding electromagnetic systems.
Secondary Coil Circuit
The secondary coil circuit plays an essential role in mutual inductance by receiving the induced voltage from changes in the primary coil. It is part of a full circuit that includes resistance, which can affect how the current flows through the circuit.

In our scenario, the secondary coil is connected to a resistor with a 12 \( \Omega \) resistance, and a 6.0 mA current is induced in it due to the primary coil's changing current.
  • The secondary coil acts as the receiver of the induced voltage, translating changes in the primary coil into a measurable current.
  • This circuit provides the pathway for the current, linking the observed voltage and current to the overall electrical circuit's properties.
Understanding the role of the secondary coil circuit is key, as it helps in analyzing how efficiently energy is transferred from the primary source to other parts of an electrical system.

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Most popular questions from this chapter

ssm A circular coil \((950 \text { turns, radius }=0.060 \mathrm{m})\) is rotating in a uniform magnetic field. At \(t=0\) s, the normal to the coil is perpendicular to the magnetic field. At \(t=0.010\) s, the normal makes an angle of \(\phi=45^{\circ}\) with the field because the coil has made one-eighth 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.

A flat coil of wire has an area \(A, N\) turns, and a resistance \(R .\) It is situated in a magnetic field, such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of \(90^{\circ},\) so that the normal becomes perpendicular to the magnetic field. The coil has an area of \(1.5 \times 10^{-3} \mathrm{m}^{2}, 50\) turns, and a resistance of 140\(\Omega .\) During the time while it is rotating, a charge of \(8.5 \times 10^{-5}\) C flows in the coil. What is the magnitude of the magnetic field?

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 \(R_{2}\) .

ssm Multiple-Concept Example 13 reviews some of the principles used in this problem. Suppose you wish to make a solenoid whose self-inductance is 1.4 \(\mathrm{mH}\) . The inductor is to have a cross-sectional area of \(1.2 \times 10^{-3} \mathrm{m}^{2}\) and a length of 0.052 \(\mathrm{m}\) . How many turns of wire are needed?

ssm The drawing shows three identical rods (A, B, and C) moving in different planes. A constant magnetic field of magnitude 0.45 T is directed along the \(+y\) axis. The length of each rod is \(L=1.3 \mathrm{m},\) and the rods each have the same speed, \(v_{\mathrm{A}}=v_{\mathrm{B}}=v_{\mathrm{C}}=2.7 \mathrm{m} / \mathrm{s}\) . For each rod, find the magnitude of the motional emf, and indicate which end (1 or 2 ) of the rod is positive.
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