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Chapter 30: Problem 6

A toroidal solenoid with mean radius \(r\) and cross-sectional area \(A\) is wound uniformly with \(N_{1}\) turns. A second toroidal solenoid with \(N_{2}\) turns is wound uniformly on top of the first, so that the two solenoids have the same cross-sectional area and mean radius. (a) What is the mutual inductance of the two solenoids? Assume that the magnetic field of the first solenoid is uniform across the cross section of the two solenoids. (b) If \(N_{1}=500\) turns, \(N_{2}=300\) turns, \(r=10.0 \mathrm{cm},\) and \(A=0.800 \mathrm{cm}^{2},\) what is the value of the mutual inductance?

Short Answer

Expert verified
The mutual inductance is 24 \, \mu \text{H}.

Step by step solution

01

Define Mutual Inductance Formula

Mutual inductance (M) between two solenoids is given by:\[ M = \frac{\mu_0 N_1 N_2 A}{2 \pi r} \]where \( \mu_0 \) is the permeability of free space, \( N_1 \) and \( N_2 \) are the number of turns in each solenoid, \( A \) is the cross-sectional area, and \( r \) is the mean radius.
02

Substitute Given Values

Substitute \( N_1 = 500 \) turns, \( N_2 = 300 \) turns, \( r = 10.0 \, \text{cm} = 0.1 \, \text{m} \), and \( A = 0.800 \, \text{cm}^2 = 0.00008 \, \text{m}^2 \) into the formula. Then use the permeability of free space \( \mu_0 = 4 \pi \times 10^{-7} \, \text{T m/A} \).
03

Calculate Mutual Inductance

Using the values in the formula:\[M = \frac{4 \pi \times 10^{-7} \times 500 \times 300 \times 0.00008}{2 \pi \times 0.1}\]which simplifies to:\[M = \frac{4 \times 300 \times 0.00008 \times 500}{2 \times 0.1} \times 10^{-7}\]Finally, calculate \( M \) by solving the above expression.
04

Final Calculation Result

The mutual inductance calculates to:\[M = 2.4 \times 10^{-5} \, \text{H}\]Therefore, the mutual inductance is \( 24 \, \mu \text{H} \).

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Key Concepts

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

Toroidal Solenoid
A toroidal solenoid is essentially a coil shaped like a doughnut, specifically designed to produce uniform magnetic fields within its interior. It has distinct properties that make it useful in electromagnetic applications, especially in circuit and magnetic field interactions.
  • **Structure:** It consists of wire wound in a coil over a circular path, usually with a circular cross-section.
  • **Properties:** The key advantage is that the magnetic field generated by toroidal solenoids is nearly entirely internal, minimizing external magnetic interference.
  • **Application:** They are used in various inductive devices and transformers due to their efficiency in containing the magnetic field.
A toroidal solenoid's unique doughnut shape ensures that the magnetic field lines are concentrated within the coil, reducing energy losses to the outside environment.
Magnetic Field
The magnetic field generated by a toroidal solenoid is a vital concept in understanding mutual inductance.
  • **Formation:** The field is created when a current passes through the coil's turns.
  • **Direction:** It forms closed loops within the solenoid鈥檚 circular path, ensuring that it stays mostly inside.
  • **Uniformity:** The central area of the toroid maintains a nearly constant magnetic field due to symmetrical winding and the closed-loop system.
The strength and uniformity of this magnetic field within a toroidal solenoid are pivotal because they influence the mutual induction between two closely wound solenoids.
Permeability of Free Space
Permeability of free space, denoted by the symbol \( \mu_0 \), measures how easily magnetic fields pass through the vacuum of space or airy environments with no material interference.
  • **Value:** It is a fundamental constant and equals \( 4 \pi \times 10^{-7} \text{ T m/A} \).
  • **Role in Inductance:** In mutual inductance calculations, \( \mu_0 \) plays an essential part because it relates the magnetic flux density to the magnetic field strength.
  • **Relevance:** It helps quantify the efficiency of magnetic field formation in the absence of magnetic materials.
Understanding the permeability of free space is crucial for calculating mutual inductance as it influences how a magnetic field propagates in an open-air setting without immediate solid material interference.
Cross-Sectional Area
The cross-sectional area \( A \) of a toroidal solenoid is the area of the circle formed by a cut perpendicular to the coiled wire.
  • **Importance:** It is a key parameter in computing inductance as it affects how much magnetic flux can be channeled through the solenoid.
  • **Relation to Inductance:** In the mutual inductance formula, the cross-sectional area directly influences the magnitude of mutual inductance by dictating the field strength interaction between the coils.
  • **Units:** Measured in square meters (\( \text{m}^2 \)), it needs to be converted from any other units (like \( \text{cm}^2 \)) for the calculation.
A larger cross-sectional area allows more magnetic field lines to pass through, enhancing the coupling and thereby increasing mutual inductance.

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Most popular questions from this chapter

In a proton accelerator used in elementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of 4.80 T. What is the magnetic-field energy in a 10.0 -cm' volume of space where \(B=4.80 \mathrm{T} ?\)

A toroidal solenoid has 500 turns, cross-sectional area \(6.25 \mathrm{cm}^{2},\) and mean radius 4.00 \(\mathrm{cm} .\) (a) Calculate the coil's self-inductance. (b) If the current decreases uniformly from 5.00 \(\mathrm{A}\) to 2.00 \(\mathrm{A}\) in 3.00 \(\mathrm{ms}\) , calculate the self- induced emf in the coil. (c) The current is directed from terminal \(a\) of the coil to terminal \(b .\) Is the direction of the induced emf from \(a\) to \(b\) or from \(b\) to \(a ?\)

Two coils are wound around the same cylindrical form, like the coils in Example \(30.1 .\) When the current in the first coil is decreasing at a rate of \(-0.242 \mathrm{A} / \mathrm{s}\) , the induced emf in the second coil has magnitude 1.65 \(\times 10^{-3} \mathrm{V}\) . (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the flux through each turn when the current in the first coil equals 1.20 \(\mathrm{A} ?\) (c) If the current in the second coil increases at a rate of \(0.360 \mathrm{A} / \mathrm{s},\) what is the magnitude of the induced emf in the first coil?

A \(7.00-\mu \mathrm{F}\) capacitor is initially charged to a potential of 16.0 \(\mathrm{V}\) . It is then connected in series with a 3.75 -mH inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?

A \(0.250-\mathrm{H}\) inductor carries a time-varying current given by the expression \(i=(124 \mathrm{mA}) \cos [(240 \pi / \mathrm{s}) t] .\) (a) Find an expression for the induced emf as a function of time. Graph the current and induced emf as functions of time for \(t=0\) to \(t=\frac{1}{60} \mathrm{s}\) (b) What is the maximum emf? What is the current when the induced emf is a maximum? (c) What is the maximum current? What is the induced emf when the current is a maximum?
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