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Chapter 29: Problem 39

A long, straight solenoid with a cross-sectional area of 8.00 \(\mathrm{cm}^{2}\) is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

Short Answer

Expert verified
The average induced emf is approximately -0.119 V.

Step by step solution

01

Understanding the Problem

We need to calculate the average induced electromotive force (emf) in a second winding encircling a solenoid when the magnetic field inside it changes. We have given values: cross-sectional area, number of turns of the solenoid per centimeter, the current, and time taken for the magnetic field to drop to zero.
02

Magnetic Field Inside the Solenoid

The magnetic field inside a long solenoid is given by the formula \( B = \mu_0 n I \), where \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}) \), \( n \) is the number of turns per meter, and \( I \) is the current. Calculate \( n \) as 90 turns/cm = 9000 turns/m. Substitute these values to find \( B \).
03

Initial Magnetic Flux

The magnetic flux through one loop of the second winding is \( \Phi = B A \), where \( A = 8.00 \ \text{cm}^2 = 8.00 \times 10^{-4} \ \text{m}^2 \) is the cross-sectional area. Calculate the initial flux using the magnetic field calculated in the previous step.
04

Change in Magnetic Flux

The magnetic field goes from its initial value (solve for \( B \) in step 2) to zero. Calculate the change in magnetic flux, \( \Delta \Phi = \Phi_{final} - \Phi_{initial} = 0 - \Phi_{initial} \).
05

Induced EMF Calculation

The average induced emf is given by Faraday's law, \( \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \), where \( N = 12 \) is the number of turns of the second winding, and \( \Delta t = 0.0400 \ s \) is the time interval. Substitute the change in magnetic flux and time to calculate the average induced emf.

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Key Concepts

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

Solenoid
A solenoid is essentially a coil of wire that is shaped into a long cylinder. When an electric current passes through its wire, it generates a magnetic field.
This is the principle behind electromagnets. Solenoids are extensively used in electromagnetics due to their capacity to create uniform magnetic fields inside their body.
Here are some important characteristics of solenoids:
  • Consist of multiple loops of wire.
  • The magnetic field inside a solenoid is strong and uniform.
  • The strength of the magnetic field (\( B \)) inside a solenoid is determined by the formula: \( B = \mu_0 n I \), where \( n \) is the number of turns per unit length, \( I \) is the current, and \( \mu_0 \) is the permeability of free space.
  • Outside the solenoid, the magnetic field is weak and can often be considered negligible.

In the context of the original exercise, the solenoid's magnetic behavior is utilized to induce an electromotive force in another coil that encircles it. This action occurs when the current in the solenoid is turned off, causing the magnetic field to collapse to zero in a fraction of a second.
Faraday's Law of Induction
Faraday's Law of Induction is a fundamental principle in electromagnetism that describes how an electric current can be induced in a loop of wire when it is exposed to a changing magnetic field.
Michael Faraday discovered that when the magnetic field around a conductor changes, it induces an electromotive force (emf) in the conductor.
Key aspects of Faraday's Law include:
  • The induced emf is equal to the negative rate of change of magnetic flux through the loop. This is mathematically expressed as \( \mathcal{E} = -\frac{d\Phi}{dt} \), where \( \Phi \) is the magnetic flux.
  • The negative sign signifies Lenz's Law, which states that the induced emf will create a current that opposes the original change in magnetic field.
  • Multi-loop systems, such as second windings around a solenoid, involve multiplying the emf by the number of turns (\( N \)) of the coil: \( \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \).

In the given exercise, Faraday's Law of Induction is applied to determine the average induced emf in the secondary loop, utilizing the change in magnetic flux from the solenoid.
Magnetic Flux
Magnetic flux is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It's a crucial concept necessary for understanding electromagnetic induction.
Magnetic flux (\( \Phi \)) is calculated as the product of the magnetic field (\( B \)) and the perpendicular area (\( A \)) it penetrates:\( \Phi = B A \).
This states that magnetic flux depends on:
  • The strength of the magnetic field: Stronger magnetic fields produce more flux.
  • The area through which the lines of magnetic field pass: Larger areas contribute to increased flux.
  • The angle between the magnetic field lines and the normal (perpendicular) to the surface: Greater angles reduce effective flux.

In the context of our exercise, the solenoid's magnetic field initially determines the magnetic flux. When this field reduces to zero as the current is switched off, a change in magnetic flux is experienced, which is pivotal in inducing emf in the second winding, as described by Faraday's Law.

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Most popular questions from this chapter

CALC In a region of space, a magnetic field points in the \(+x\) -direction (toward the right). Its magnitude varies with position according to the formula \(B_{x}=B_{0}+b x,\) where \(B_{0}\) and \(b\) are positive constants, for \(x \geq 0 .\) A flat coil of area \(A\) moves with uniform speed \(v\) from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

A rectangle measuring 30.0 \(\mathrm{cm}\) by 40.0 \(\mathrm{cm}\) is located inside a region of a spatially uniform magnetic field of 1.25 \(\mathrm{T}\) , with the field perpendicular to the plane of the coil (Fig. E29.24). The coil is pulled out at a steady rate of 2.00 \(\mathrm{cm} / \mathrm{s}\) traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field; (b) partly inside the field; (c) all outside the field.

The compound \(\mathrm{SiV}_{3}\) is a type-II superconductor. At temperatures near absolute zero the two critical fields are \(B_{\mathrm{cl}}=55.0 \mathrm{mT}\) and \(B_{\mathrm{c} 2}=15.0 \mathrm{T}\) . The normal phase of \(\mathrm{Si} \mathrm{V}_{3}\) has a magnetic susceptibility close to zero. A long, thin \(\mathrm{SiV}_{3}\) cylinder has its axis parallel to an external magnetic field \(\vec{\boldsymbol{B}}_{0}\) in the \(+x\) -direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the \(x\) -axis. At a temperature near absolute zero, the external magnetic field is slowly increased from zero. What are the resultant magnetic field \(\vec{\boldsymbol{B}}\) and the magnetization \(\vec{M}\) inside the cylinder at points far from its ends (a) just before the magnetic flux begins to penetrate the material, and (b) just after the material becomes completely normal?

Are Motional emfs a Practical Source of Electricity? How fast (in \(\mathrm{m} / \mathrm{s}\) and mph) would a 5.00 -cm copper bar have to move at right angles to a \(0.650-\mathrm{T}\) magnetic field to generate 1.50 \(\mathrm{V}\) (the same as a A \(\mathrm{A}\) battery) across its ends? Does this seem like a practical way to generate electricity?

CALC Displacement Current in a Wire. A long, straight, copper wire with a circular cross-sectional area of 2.1 \(\mathrm{mm}^{2}\) carries a current of 16 \(\mathrm{A}\) . The resistivity of the material is \(2.0 \times\) \(10^{-8} \Omega \cdot \mathrm{m} .\) (a) What is the uniform electric field in the material? (b) If the current is changing at the rate of \(4000 \mathrm{A} / \mathrm{s},\) at what rate is the electric field in the material changing? (c) What is the displacement current density in the material in part (b)? (Hint: Since \(K\) for copper is very close to \(1,\) use \(\epsilon=\epsilon_{0.2}\) (d) If the current is changing as in part (b), what is the magnitude of the magnetic field 6.0 cm from the center of the wire? Note that both the conduction current and the displacement current should be included in the calculation of \(B .\) Is the contribution from the displacement current significant?
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