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Chapter 30: Problem 3

Two toroidal solenoids are wound around the same form so that the magnetic ficld of one passes through the turns of the other. Solenoid 1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1 is \(6.52 \mathrm{~A},\) the average flux through each turn of solenoid 2 is \(0.0320 \mathrm{~Wb}\). (a) What is the mutual inductance of the pair of solcnoids? (b) When the current in solcnoid 2 is 2.54 A. what is the average flux through each turn of solenoid \(1 ?\)

Short Answer

Expert verified
The mutual inductance of the pair of solenoids is 0.00491 H. When the current in Solenoid 2 is 2.54 A, the average flux through each turn of Solenoid 1 is approximately 0.0124854 Wb.

Step by step solution

01

Find the mutual inductance

The mutual inductance 'M' can be determined by rearranging the equation \(M = \frac{\Phi_2}{I_1}\) where \(\Phi_2 = 0.0320\) Wb is the flux through Solenoid 2 and \(I_1 = 6.52\) A is the current through Solenoid 1. Substituting the given values we find \(M = \frac{0.0320}{6.52} = 0.00491\) H.
02

Determine the average flux through Solenoid 1

By using the formula \(\Phi_1 = MI_2\) we can calculate the average flux passing through Solenoid 1. This involves substituting \(M = 0.00491\) H from Step 1 and the given \(I_2 = 2.54\) A, giving \(\Phi_1 = 0.00491 \times 2.54 = 0.0124854\) Wb

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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 a type of solenoid that is shaped like a doughnut or a ring. This design is wrapped tightly with coiled wire and typically used in applications that require a strong, uniform magnetic field inside the coil. Unlike a traditional straight solenoid, the toroidal configuration ensures that the magnetic field lines are contained within the coil itself.

This means less energy is lost to the surrounding space, making toroidal solenoids efficient and effective for circuits needing magnetic coupling.

They are often used in electrical devices such as transformers and inductors, where space is limited but efficient magnetic field use is critical. The toroidal solenoid in the exercise has two sets of coils wound around the same form, allowing for interactions that lead to mutual inductance.
Magnetic Flux
Magnetic flux is a measure of how much magnetic field passes through a given area. It is calculated as the product of the magnetic field strength and the area perpendicular to the field through which the lines of force pass.

In simpler terms, it quantifies the total magnetic field that threads through a specific area.

The formula for magnetic flux is given by \[\Phi = B \cdot A \cdot \cos(\theta)\]where
  • \(B\) is the magnetic field strength,
  • \(A\) is the area, and
  • \(\theta\) is the angle between the magnetic field and the perpendicular to the area.

In the exercise, magnetic flux is used to understand how the magnetic fields from one solenoid pass through another. This is crucial for calculating mutual inductance, which relies on the amount of magnetic flux one coil creates in the other.
Electromagnetic Induction
Electromagnetic induction is the process through which a change in magnetic flux generates an electric current within a conductor. Discovered by Michael Faraday, this principle is foundational in creating electric power from magnetic fields.

When a conductor, like a coil of wire, experiences a change in magnetic flux, an electromotive force (EMF) is induced. This can either generate electricity or change the current in a nearby circuit. The extent of induction is not only dependent on the change in the magnetic field but also on the number of turns in the coil and the speed of change.

In this exercise, electromagnetic induction comes into play when changes in the current of one solenoid affect the magnetic flux through another. Since the coils are wound on the same form, the magnetic field from each coil affects the other, illustrating the principle of mutual inductance.

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Most popular questions from this chapter

An inductor used in a de power supply has an inductance of \(12.0 \mathrm{H}\) and a resistance of \(180 \Omega\). It carries a current of \(0.500 \mathrm{~A}\). (a) What is the energy stored in the magnetic field? (b) At what rate is thermal energy developed in the inductor? (c) Does your answer to part (b) mean that the magnetic-field energy is decreasing with time? Explain.

An air-filled toroidal solenoid has a mean radius of \(15.0 \mathrm{~cm}\) and a cross-sectional area of \(5.00 \mathrm{~cm}^{2}\). When the current is \(12.0 \mathrm{~A}\), the encrgy stored is \(0.390 \mathrm{~J}\). How many turns does the winding have?

CALC A coil has 400 turns and self-inductance \(7.50 \mathrm{mH}\) The current in the coil varies with time according to \(i=\) \((680 \mathrm{~mA}) \cos (\pi t / 0.0250 \mathrm{~s}) .\) (a) What is the maximum emf induced in the coil? (b) What is the maximum average flux through each turn of the coil? (c) At \(t=0.0180 \mathrm{~s}\), what is the magnitude of the induced emf?

Inductance of a Solenoid. (a) A long, straight solenoid has \(N\) turns, uniform cross-sectional area \(A,\) and length \(1 .\) Show that the inductance of this solenoid is given by the equation \(L=\mu_{0} A N^{2} / l\) Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because \(B\) is actually smaller at the ends than at the center. For this reason, your answer is actually an upper limit on the inductance.) (b) \(A\) metallic laboratory spring is typically \(5.00 \mathrm{~cm}\) long and \(0.150 \mathrm{~cm}\) in diameter and has 50 coils. If you connect such a spring in an clectric circuit, how much self-inductance must you include for it if you model it as an idcal solcnoid?

I-C Oscillations. A capacitor with capacitance \(6.00 \times 10^{-5} \mathrm{~F}\) is charged by connecting it to a \(12.0 \mathrm{~V}\) battery. The capacitor is disconnected from the battery and connected across an inductor with \(L=1.50 \mathrm{H}\). (a) What are the angular frequency \(\omega\) of the electrical os cillations and the period of these ascillations (the time for one oscillation)? (b) What is the initial charge on the capacitor? (c) Ilow much energy is initially stored in the capacitor? (d) What is the charge on the capacitor \(0.0230 \mathrm{~s}\) after the connection to the inductor is made? Interpret the sign of your answer. (c) At the time given in part (d), what is the current in the inductor? Interpret the sign of your answer. (f) At the time given in part (d), how much electrical energy is stored in the capacitor and how much is stored in the inductor?
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