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Chapter 30: Problem 21

A stiff wire bent into a semicircle of radius \(a=2.0 \mathrm{~cm}\) is rotated at constant angular speed 40 rev/s in a uniform \(20 \mathrm{mT}\) magnetic field. What are the (a) frequency and (b) amplitude of the emf induced in the loop?

Short Answer

Expert verified
(a) Frequency: 40 Hz, (b) Amplitude of emf: 3.14 mV.

Step by step solution

01

Determine the Frequency

The frequency of the induced electromotive force (emf) is the same as the frequency of rotation. Since the wire is rotating at 40 revolutions per second, the frequency \( f \) is simply \( f = 40 \text{ Hz} \).
02

Calculate the Magnetic Flux

The magnetic flux \( \Phi \) through the loop is given by \( \Phi = B \cdot A \), where \( B \) is the magnetic field and \( A \) is the area of the semicircle. The area of a full circle is \( \pi a^2 \), thus the area of the semicircle is \( A = \frac{1}{2} \pi a^2 = \frac{1}{2} \cdot \pi \cdot (0.02)^2 \approx 6.28 \times 10^{-4} \; \text{m}^2 \).
03

Write the Induced Emf

The induced emf (\( \varepsilon \)) can be found using Faraday's Law of Induction, which states \( \varepsilon = \frac{d\Phi}{dt} \). Since the semicircle is rotating with constant angular speed \( \omega = 2\pi f \), the emf can be expressed as \( \varepsilon = B \omega A \).
04

Calculate the Amplitude of the Induced Emf

Substitute the known values into the expression from Step 3: \( B = 20 \times 10^{-3} \; T \), \( \omega = 2\pi \times 40 \; \text{rad/s} \), and \( A \approx 6.28 \times 10^{-4} \; \text{m}^2 \). So, calcualte the amplitude of the emf:\[\varepsilon = 0.02 \times 2\pi \times 40 \times 6.28 \times 10^{-4} \approx 0.00314 \; \text{V} \].

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Key Concepts

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

Faraday's Law
Faraday's Law of electromagnetic induction gives us a clear understanding of how voltage can be created in a circuit by a changing magnetic field. This is crucial in understanding how the induced emf in a wire loop or coil works. The law states that the induced emf in a closed loop is equal to the negative rate of change of the magnetic flux through the loop. More simply,
  • When the magnetic environment around a coil of wire changes, it induces a current in the wire.
  • The formula is given by: \( \varepsilon = -\frac{d\Phi}{dt} \), where \( \varepsilon \) is the induced emf and \( \Phi \) is the magnetic flux.
  • The negative sign represents Lenz's law, indicating that the induced emf generates a current that opposes the change in the magnetic flux.
This relationship helps us calculate quantities such as voltage in systems where the magnetic field's intensity changes over time.
Magnetic Flux
Magnetic flux refers to the total magnetic field that passes through a given area. For the semicircular wire loop in the problem, we need to calculate how much of the magnetic field interacts with it.
  • The formula for magnetic flux is \( \Phi = B \cdot A \cdot \cos \theta \), where \( B \) is the magnetic field strength, \( A \) is the area, and \( \theta \) is the angle between the field and the normal to the surface.
  • In the exercise, the angle \( \theta \) is 0 degrees since the field is perpendicular to the plane of the loop, meaning \( \cos 0 = 1 \).
  • The semicircle area calculation is done using \( A = \frac{1}{2} \pi a^2 \), which simplifies the determination of magnetic flux across the loop.
This concept helps visualize how a magnetic field's strength and orientation relative to the loop influences the resultant magnetic flux.
Angular Speed
Angular speed is a measure of how quickly something rotates. In the context of electromagnetic induction, this dictates how fast the wire cuts through magnetic field lines and impacts the rate of change of flux.
  • Angular speed \( \omega \) is often given in radians per second and is related to frequency \( f \) by \( \omega = 2\pi f \).
  • In this problem, the wire rotates at 40 revolutions per second, which means its angular speed is \( \omega = 2\pi \times 40 \) rad/s.
  • This constant angular speed ensures that the induced emf is also periodic and predictable, mirroring the regular motion of the wire.
Understanding angular speed is vital because it directly affects how much emf is induced in systems involving rotating components, such as generators.
Induced Emf
Induced electromotive force (emf) is a key outcome of electromagnetic induction. In the exercise, it is generated when a rotating semicircular wire moves through a magnetic field.
  • Using Faraday's law, the induced emf can be computed from the rate of change of magnetic flux: \( \varepsilon = B \omega A \), where \( B \) is the magnetic field, \( \omega \) is the angular speed, and \( A \) is the area of the semicircle.
  • This formula reflects how rapidly the loop cuts magnetic flux lines, thus inducing a voltage across it.
  • The calculation leads to an amplitude of \( \varepsilon \approx 0.00314 \text{ V} \) in this specific scenario, derived from substituting the given values.
This computation demonstrates how induced emf varies with the strength of the magnetic field, the speed of rotation, and the loop's geometry.

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Most popular questions from this chapter

A coil \(C\) of \(N\) turns is placed around a long solenoid \(\mathrm{S}\) of radius \(R\) and \(n\) turns per unit length, as in Fig. 30-67. (a) Show that the mutual in. ductance for the coil-solenoid combination is given by \(M=\mu_{0} \pi R^{2} n N\). (b) Explain why \(M\) does not depend on the shape, size, or possible lack of close packing of the coil.

Two straight conducting rails form a right angle. A conducting bar in contact with the rails starts at the vertex at time \(t=0\) and moves with a constant velocity of \(5.20 \mathrm{~m} / \mathrm{s}\) along them. A magnetic field with \(B=0.350 \mathrm{~T}\) is directed out of the page. Calculate (a) the flux through the triangle formed by the rails and bar at \(t=3.00 \mathrm{~s}\) and \((\mathrm{b})\) the emf around the triangle at that time. (c) If the emf is \(\mathscr{8}=a t^{n}\), where \(a\) and \(n\) are constants, what is the value of \(n ?\)

One hundred turns of (insulated) copper wire are wrapped around a wooden cylindrical core of cross-sectional area \(1.20 \times 10^{-3} \mathrm{~m}^{2}\). The two ends of the wire are connected to a resistor. The total resistance in the circuit is \(13.0 \Omega\). If an externally applied uniform longitudinal magnetic field in the core changes from \(1.60 \mathrm{~T}\) in one direction to \(1.60 \mathrm{~T}\) in the opposite direction, how much charge flows through a point in the circuit during the change?

A square loop of wire is held in a uniform \(0.24 \mathrm{~T}\) magnetic field directed perpendicular to the plane of the loop. The length of each side of the square is decreasing at a constant rate of \(5.0 \mathrm{~cm} / \mathrm{s}\). What emf is induced in the loop when the length is \(12 \mathrm{~cm}\) ?

A coil is connected in series with a \(10.0 \mathrm{k} \Omega\) resistor. An ideal \(50.0 \mathrm{~V}\) battery is applied across the two devices, and the current reaches a value of \(2.00 \mathrm{~m}\) A after \(5.00 \mathrm{~ms}\). (a) Find the inductance of the coil. (b) How much energy is stored in the coil at this same moment?
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