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Chapter 29: Problem 1

A single loop of wire with an area of 0.0900 \(\mathrm{m}^{2}\) is in a uniform magnetic field that has an initial value of 3.80 \(\mathrm{T}\) , is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 \(\mathrm{T} / \mathrm{s}\) (a) What emf is induced in this loop? (b) If the loop has a resistance of \(0.600 \Omega,\) find the current induced in the loop.

Short Answer

Expert verified
The induced emf is 0.0171 V, and the induced current is 0.0285 A.

Step by step solution

01

Understanding Faraday's Law of Induction

The electromotive force (emf) induced in a loop is given by Faraday's Law, \( \varepsilon = - \frac{d\Phi}{dt} \), where \( \Phi \) is the magnetic flux through the loop. The magnetic flux \( \Phi = B \cdot A \), where \( B \) is the magnetic field and \( A \) is the area of the loop.
02

Calculating the Change in Magnetic Flux

The magnetic flux \( \Phi = B \times A \). So, \( \Phi = 3.80 \text{ T} \times 0.0900 \text{ m}^2 = 0.342 \text{ Wb} \). The rate of change of the magnetic field is \( \frac{dB}{dt} = -0.190 \text{ T/s} \), and hence \( \frac{d\Phi}{dt} = A \cdot \frac{dB}{dt} = 0.0900 \cdot (-0.190) \).
03

Calculating the Induced Emf

Using \( \frac{d\Phi}{dt} = 0.0900 \cdot (-0.190) = -0.0171 \text{ Wb/s} \), Faraday's law gives \( \varepsilon = - \frac{d\Phi}{dt} = 0.0171 \text{ V} \). So, the induced emf in the loop is \( 0.0171 \text{ V} \).
04

Applying Ohm's Law to Find Induced Current

Ohm's Law states \( I = \frac{\varepsilon}{R} \), where \( I \) is the induced current, \( \varepsilon = 0.0171 \text{ V} \), and \( R = 0.600 \Omega \). Substituting these values, we find \( I = \frac{0.0171}{0.600} \).
05

Calculating the Induced Current

From the previous step, \( I = \frac{0.0171}{0.600} = 0.0285 \text{ A} \). Therefore, the induced current in the loop is \( 0.0285 \text{ A} \).

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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
Faraday's Law of Induction is a fundamental principle in electromagnetism that explains how an electromotive force (emf) is generated in a conductor when it experiences a change in magnetic flux. The law is mathematically expressed as \[\varepsilon = - \frac{d\Phi}{dt}\]where
  • \( \varepsilon \) is the induced emf,
  • \( \Phi \) is the magnetic flux, and
  • \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux over time.
The negative sign in the formula is in accordance with Lenz's Law. This law states that the direction of the induced emf opposes the change in flux that produced it. To make sense of this, imagine a loop of wire in a magnetic field. If the magnetic field within the loop changes, perhaps by getting stronger, weaker, or changing direction, this change will induce an emf in the loop, prompting an electric current. This induced current, by nature, acts to oppose the original change in flux by generating its own magnetic field.
Ohm's Law
Ohm's Law is a simple and essential formula in the field of electronics and electrical engineering. It relates three key quantities: voltage (\(V\)), current (\(I\)), and resistance (\(R\)). The law is defined as:\[I = \frac{V}{R}\]This means that the current \(I\) flowing through a conductor between two points is directly proportional to the voltage \(V\) across the two points and inversely proportional to the resistance \(R\) of the conductor. Ohm鈥檚 Law helps us predict how much current will flow through a circuit given a certain resistance and voltage. In the context of electromagnetic induction, once the induced emf is calculated with Faraday's Law, Ohm's Law can then be used to determine the induced current.
Magnetic Flux
Magnetic flux is a vital concept for understanding electromagnetic induction. It describes the total magnetic field (\(B\)) passing through an area (\(A\)). Mathematically, magnetic flux \( \Phi \) is defined as:\[\Phi = B \cdot A \cdot \cos\theta\]where
  • \(B\) is the magnetic field strength,
  • \(A\) is the area the magnetic field lines pass through, and
  • \(\theta\) is the angle between the magnetic field and the perpendicular to the surface
When the field is perpendicular to the surface, \(\cos\theta = 1\), maximizing the flux, as in the exercise scenario. If the magnetic field changes over time, either in strength or direction, the magnetic flux changes as well. This change results in an induced emf and hence, an electrical current, as demonstrated in Faraday's Law.

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Most popular questions from this chapter

CALC A dielectric of permittivity \(3.5 \times 10^{-11} \mathrm{F} / \mathrm{m}\) completely fills the volume between two capacitor plates. For \(t>0\) the electric flux through the dielectric is \(\left(8.0 \times 10^{3} \mathrm{V} \cdot \mathrm{s} / \mathrm{s}^{3}\right) t^{3}\) . The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal 21\(\mu \mathrm{A} ?\)

A long, thin solenoid has 900 turns per meter and radius 2.50 \(\mathrm{cm} .\) The current in the solenoid is increasing at a uniform rate of 60.0 \(\mathrm{A} / \mathrm{s}\) . What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 \(\mathrm{cm}\) from the axis of the solenoid; (b) 1.00 \(\mathrm{cm}\) from the axis of the solenoid?

Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earth's magnetic field at your location is horizontal and has magnitude \(8.0 \times 10^{-5} \mathrm{T},\) and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 \(\mathrm{V}\) and estimate that you can rotate the coil at 30 \(\mathrm{rpm}\) by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000 . (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you think this device is feasible? Explain.

CALC A conducting rod with length \(L=0.200 \mathrm{m},\) mass \(m=0.120 \mathrm{kg},\) and resistance \(R=80.0 \Omega\) moves without friction on metal rails as shown in Fig. \(29.11 .\) A uniform magnetic field with magnitude \(B=1.50 \mathrm{T}\) is directed into the plane of the figure. The rod is initially at rest, and then a constant force with magnitude \(F=1.90 \mathrm{N}\) and directed to the right is applied to the bar. How many seconds after the force is applied does the bar reach a speed of \(25,0 \mathrm{m} / \mathrm{s} ?\)

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?
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