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Chapter 31: Problem 35

A coil of area \(0.100 \mathrm{m}^{2}\) is rotating at \(60.0 \mathrm{rev} / \mathrm{s}\) with the axis of rotation perpendicular to a \(0.200-\mathrm{T}\) magnetic field. (a) If the coil has 1000 turns, what is the maximum emf generated in it? (b) What is the orientation of the coil with respect to the magnetic field when the maximum induced voltage occurs?

Short Answer

Expert verified
The maximum emf generated in the coil will be 75.40 V. The orientation of the coil with respect to the magnetic field when the maximum induced voltage occurs is such that the plane of the coil is parallel to the magnetic field.

Step by step solution

01

Calculation of angular velocity

Coil is rotating at 60.0 rev/s. Angular velocity is given by \(\omega = 2\pi \cdot\)frequency. So, replace frequency with 60.0 rev/s and calculate \(\omega\). The result is \(\omega = 2\pi \cdot 60.0 = 376.99 \, \text{s}^{-1}\).
02

Calculation of maximum emf

Substitute N=1000 turns, A=0.100 m虏, B=0.200 T, and \(\omega = 376.99 \, \text{s}^{-1}\) into \(\varepsilon = NAB\omega \sin{\theta}\) and calculate \(\varepsilon\). The result is \(\varepsilon = 1000 \cdot 0.100 \cdot 0.200 \cdot 376.99 \cdot \sin{90^\circ} = 75.40 \, V\). Recall that \(\sin{90^\circ} = 1\).
03

Statement about orientation of coil

When the maximum induced voltage occurs, the coil is perpendicular to the magnetic field. In other words, the plane of the coil is parallel to the magnetic field.

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Key Concepts

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

Faraday's Law of Induction
Electromagnetic induction describes the process by which a changing magnetic field within a coil of wire generates electricity.
Faraday's Law of Induction is a fundamental principle that defines how this process occurs. According to this law, the induced electromotive force (emf) in a closed loop is directly proportional to the rate of change of the magnetic flux through the loop.

The equation representing Faraday's Law is given as:
  • \( \varepsilon = -N \frac{d\Phi}{dt} \)
Here, \( \varepsilon \) is the induced emf, \( N \) is the number of turns in the coil, and \( \frac{d\Phi}{dt} \) represents the rate of change of magnetic flux.
The negative sign signifies Lenz's Law, which indicates that the induced current's direction will oppose the change in magnetic flux.

In our original exercise, we see this law applied to determine the maximum emf in a rotating coil within a magnetic field.
Magnetic Field
A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts.
The strength of a magnetic field is measured in teslas (T).
In the exercise, a magnetic field of 0.200 T interacts with the coil. This interaction is crucial for inducing emf according to Faraday's Law.

When a coil rotates within a magnetic field, the angle between the coil and the magnetic field changes continuously.
  • At certain angles, particularly when the coil is perpendicular to the magnetic field lines, the rate of change of magnetic flux is maximized鈥攔esulting in maximum emf generation.

In the exercise, the maximum emf occurs when the plane of the coil aligns with the direction of the magnetic field, specifically when their orientation is perpendicular.
Angular Velocity
Angular velocity is a measure of the rate of rotation, it describes how fast an object rotates about an axis.
It is usually expressed in radians per second (rad/s).
The relationship between angular velocity and frequency of rotation is given by:
  • \( \omega = 2\pi \cdot \text{frequency} \)

In our example, a coil rotates at 60 revolutions per second. Applying the formula, the angular velocity \( \omega \) becomes \( 376.99 \, \text{s}^{-1} \).
This calculated angular velocity plays a vital role in determining the maximum emf the coil generates, as it is inserted into the equation \( \varepsilon = NAB\omega \sin \theta \).
In scenarios involving electromagnetic induction, faster rotations generally lead to higher values of emf.
Emf (Electromotive Force)
Electromotive force (emf) is a term used to describe the potential difference generated in an electric circuit.
It is measured in volts (V), and although referred to as "force," it actually represents energy per unit charge.
When a coil moves through a magnetic field, electrical energy is created. This induced emf can drive an electric current around the circuit, making it fundamental in devices like electric generators.

The equation for induced emf, derived from Faraday's Law, is:
  • \( \varepsilon = NAB\omega \sin \theta \)
In this context:
  • \( N \) is the number of turns in the coil.
  • \( A \) is the area of the coil.
  • \( B \) is the magnetic field strength.
  • \( \omega \) is the angular velocity.
  • \( \theta \) is the angle between the normal to the coil and the magnetic field.

In the example solution, inserting the values leads to finding a maximum emf of 75.40 V when \( \theta = 90^\circ \), where \( \sin 90^\circ = 1 \).
This setup of these variables can yield the highest possible emf in a rotating coil under the given conditions.

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Most popular questions from this chapter

A rectangular loop of area \(A\) is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to \(B=B_{\max } e^{-l / \tau},\) where \(B_{\max }\) and \(\tau\) are constants. The field has the constant value \(B_{\max }\) for \(t<0\) (a) Use Faraday's law to show that the emf induced in the loop is given by $$\mathcal{E}=\frac{A B_{\max }}{\tau} e^{-t / \tau}$$ (b) Obtain a numerical value for \(\boldsymbol{\varepsilon}\) at \(t=4.00 \mathrm{s}\) when \(A=0.160 \mathrm{m}^{2}, B_{\max }=0.350 \mathrm{T},\) and \(\tau=2.00 \mathrm{s} .\) (c) For the values of \(A, B_{\max },\) and \(\tau\) given in (b), what is the maximum value of \(\mathcal{E} ?\)

Magnetic field values are often determined by using a device known as a search coil. This technique depends on the measurement of the total charge passing through a coil in a time interval during which the magnetic flux linking the windings changes either because of the motion of the coil or because of a change in the value of \(B\) (a) Show that as the flux through the coil changes from \(\Phi_{1}\) to \(\Phi_{2},\) the charge transferred through the coil will be given by \(Q=N\left(\Phi_{2}-\Phi_{1}\right) / R,\) where \(R\) is the resistance of the coil and a sensitive ammeter connected across it and \(N\) is the number of turns. (b) As a specific example, calculate \(B\) when a 100 -turn coil of resistance \(200 \Omega\) and crosssectional area \(40.0 \mathrm{cm}^{2}\) produces the following results. A total charge of \(5.00 \times 10^{-4} \mathrm{C}\) passes through the coil when it is rotated in a uniform field from a position where the plane of the coil is perpendicular to the field to a position where the coil's plane is parallel to the field.

In a 250 -turn automobile alternator, the magnetic flux in each turn is \(\Phi_{B}=\left(2.50 \times 10^{-4} \mathrm{Wb}\right) \cos (\omega t),\) where \(\omega\) is the angular speed of the alternator. The alternator is geared to rotate three times for each engine revolution. When the engine is running at an angular speed of 1000 rev/min, determine (a) the induced emf in the alternator as a function of time and (b) the maximum emf in the alternator.

Very large magnetic fields can be produced using a procedure called flux compression. A metallic cylindrical tube of radius \(R\) is placed coaxially in a long solenoid of somewhat larger radius. The space between the tube and the solenoid is filled with a highly explosive material. When the explosive is set off, it collapses the tube to a cylinder of radius \(r

A solenoid wound with 2000 turns/m is supplied with current that varies in time according to \(I=\) (4A) \(\sin (120 \pi t),\) where \(t\) is in seconds. A small coaxial circular coil of 40 turns and radius \(r=5.00 \mathrm{cm}\) is located inside the solenoid near its center. (a) Derive an expression that describes the manner in which the emf in the small coil varies in time. (b) At what average rate is energy delivered to the small coil if the windings have a total resistance of \(8.00 \Omega ?\)
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