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Chapter 22: Problem 56

A long, current-carrying solenoid with an air core has 1750 turns per meter of length and a radius of 0.0180 \(\mathrm{m}\) . A coil of 125 turns is wrapped tightly around the outside of the solenoid, so it has virtually the same radius as the solenoid. What is the mutual inductance of this system?

Short Answer

Expert verified
The mutual inductance is approximately \( 3.52 \times 10^{-4} \, \text{H} \).

Step by step solution

01

Understanding Mutual Inductance

The mutual inductance, \( M \), between a solenoid and a wire coil is given by the formula: \[ M = \mu_0 \cdot n_1 \cdot n_2 \cdot V \] where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot \text{m/A} \) is the permeability of free space, \( n_1 \) is the number of turns per unit length of the solenoid, \( n_2 \) is the total number of turns in the coil, and \( V \) is the volume of the solenoid.
02

Calculate the Volume of the Solenoid

The volume, \( V \), of the cylindrical solenoid is calculated as: \[ V = A \cdot L \] where \( A = \pi r^2 \) is the cross-sectional area of the solenoid and \( L \) is its length. However, as \( M \) is independent of \( L \), we just need the area:\[ A = \pi (0.0180 \, \text{m})^2 \approx 1.0179 \times 10^{-3} \, \text{m}^2 \]
03

Insert Values into the Formula

Now substitute the known values into the mutual inductance formula: \[ M = \mu_0 \cdot n_1 \cdot n_2 \cdot A \] Substituting,\[ M = 4\pi \times 10^{-7} \, \text{T}\cdot \text{m/A} \times 1750 \, \text{turns/m} \times 125 \, \text{turns} \times 1.0179 \times 10^{-3} \, \text{m}^2 \]
04

Perform the Calculations

Calculate using the substituted values:First, compute the product:\[ M = 4\pi \times 10^{-7} \times 1750 \times 125 \times 1.0179 \times 10^{-3} \]Then, simplify:\[ M \approx 4\pi \times 10^{-7} \times 2.2303125\cdot10^{-1} \] \[ M \approx 3.52 \times 10^{-4} \, \text{H} \]
05

Conclusion

The mutual inductance of the system is approximately \( 3.52 \times 10^{-4} \, \text{H} \).

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Key Concepts

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

Understanding Solenoid
A solenoid is a coil of wire designed to create a magnetic field when an electric current passes through it. It is often used in electromagnetic applications because of its simple construction and effectiveness in producing a uniform and controllable magnetic field. Solenoids are generally made by wrapping the wire in a helix shape around a cylindrical form.

When a current flows through the solenoid, it produces a magnetic field inside it, which resembles the field created by a bar magnet. This magnetic field is strong and uniform at the center while weakening towards the edges. The strength of this field depends on factors such as the number of turns of the coil, the current flowing through the coil, and the medium within the solenoid, which is often air or another non-conductive material.
Understanding Turns Per Meter
Turns per meter is a term used to describe the density of the winding of coil within a solenoid. It indicates how many wire loops are present in a specific length of the solenoid, usually measured in turns per meter (turns/m). This measure is crucial because it directly affects the magnetic field strength inside the solenoid.

A higher number of turns per meter generally indicates a stronger magnetic field, assuming that the current in the coil remains constant. It is a critical factor in calculating mutual inductance, as it contributes to the effectiveness of the coil in inducing magnetic fields in nearby coils or conductors. For example, in the given exercise, a solenoid with 1750 turns per meter was used to determine mutual inductance.
Permeability of Free Space
The permeability of free space, denoted as \( \mu_0 \), is a fundamental physical constant that characterizes the ability of a vacuum to support the formation of a magnetic field. It is a measure of how well a medium allows magnetic field lines to pass through it. In the international system of units, it is given the value of \( 4\pi \times 10^{-7} \) Tesla meter per Ampere (T m/A).

This constant is significant in physics and engineering because it provides a baseline for calculating magnetic fields in free space, allowing comparisons with different materials. For systems like solenoids, the permeability of free space is essential in determining parameters like mutual inductance, as it affects the strength and extent of the magnetic interaction between conductors.
Characteristics of a Cylindrical Solenoid
A cylindrical solenoid is a common type of solenoid with a tube-like shape. This design is advantageous because it concentrates the magnetic field inside the cylinder, making it intense and relatively uniform. The cylindrical solenoid is usually defined by its length, radius, and the thickness of the coil wound around it.

In practical applications, the cylindrical solenoid's benefits include its straightforward design, ease of manufacturing, and efficiency in magnetics applications. When calculating mutual inductance, the shape of the solenoid impacts the calculations, especially concerning its radius, which is needed to determine the cross-sectional area of the solenoid.

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Most popular questions from this chapter

A generator is connected across the primary coil ( \(N_{p}\), 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 \(\mathrm{Ohm}^{\prime}\) s law as applied to the secondary coil.

A motor is designed to operate on 117 \(\mathrm{V}\) and draws a current of 12.2 \(\mathrm{A}\) when it first starts up. At its normal operating speed, the motor draws a current of 2.30 A. Obtain (a) the resistance of the armature coil, (b) the back emf developed at normal speed, and (c) the current drawn by the motor at one-third of the normal speed.

Mutual induction can be used as the basis for a metal detector. A typical setup uses two large coils that are parallel to each other and have a common axis. Because of mutual induction, the ac generator connected to the primary coil causes an emf of 0.46 V to be induced in the secondary coil. When someone without metal objects walks through the coils, the mutual inductance and, thus, the induced emf do not change much. But when a person carrying a handgun walks through, the mutual inductance increases. The change in emf can be used to trigger an alarm. If the mutual inductance increases by a factor of three, find the new value of the induced emf.

ssm The plane of a flat, circular loop of wire is horizontal. An external magnetic field is directed perpendicular to the plane of the loop. The magnitude of the external magnetic field is increasing with time. Because of this increasing magnetic field, an induced current is flowing clockwise in the loop, as viewed from above. What is the direction of the external magnetic field? Justify your conclusion.

Multiple-Concept Example 13 reviews the concepts used in this problem. A long solenoid (cross-sectional area \(=1.0 \times 10^{-6} \mathrm{m}^{2}\) , number of turns per unit length \(=2400\) turns/m) is bent into a circular shape so it looks like a donut. This wire-wound donut is called a toroid. Assume that the diameter of the solenoid is small compared to the radius of the toroid, which is 0.050 \(\mathrm{m}\) . Find the emf induced in the toroid when the current decreases to 1.1 A from 2.5 A in a time of 0.15 s.
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