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Chapter 29: Problem 5

A circular loop of wire with a radius of 12.0 \(\mathrm{cm}\) and oriented in the horizontal \(x y\) -plane is located in a region of uniform magnetic field. A field of 1.5 \(\mathrm{T}\) is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 \(\mathrm{ms}\) , find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Short Answer

Expert verified
(a) 34.05 V; (b) Counterclockwise.

Step by step solution

01

Calculate the Initial Magnetic Flux

The initial magnetic flux \( \Phi_i \) through the loop can be calculated using the formula: \[ \Phi_i = B \times A = B \times \pi r^2 \] where \( B = 1.5 \mathrm{T} \) and \( r = 0.12 \mathrm{m} \). Substitute the values: \[ \Phi_i = 1.5 \times \pi \times (0.12)^2 \approx 0.0681 \mathrm{Wb} \] (Weber).
02

Determine the Final Magnetic Flux

When the loop is removed from the magnetic field, the final magnetic flux \( \Phi_f \) is 0 since there is no magnetic field acting on the loop. Hence, \( \Phi_f = 0 \mathrm{Wb} \).
03

Calculate the Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is given by \( \Delta \Phi = \Phi_f - \Phi_i \). Substitute the values: \[ \Delta \Phi = 0 - 0.0681 = -0.0681 \mathrm{Wb} \].
04

Calculate the Average Induced EMF

Using Faraday's Law of Induction, the average induced EMF \( \mathcal{E} \) is given by: \[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \] Here, \( \Delta t = 2.0 \times 10^{-3} \mathrm{s} \). Substitute the values and calculate: \[ \mathcal{E} = -\frac{-0.0681}{2.0 \times 10^{-3}} \approx 34.05 \mathrm{V} \]. Thus, the average induced EMF is approximately \( 34.05 \mathrm{V} \).
05

Determine the Direction of Induced Current

According to Lenz's Law, the induced current will oppose the change in flux. Since the magnetic field is directed upward and is being removed, the induced current will create a magnetic field upward to oppose this effect. Viewed from above, a counterclockwise current is needed to create an upward magnetic field. Therefore, the induced current is counterclockwise.

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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. It states that a change in magnetic flux through a loop of wire will induce an electromotive force (EMF) in the wire. This change in magnetic flux can occur due to a changing magnetic field, the movement of the wire, or both. The formula representing Faraday's Law is given by:
  • \( \mathcal{E} = -\frac{d\Phi}{dt} \)
Here, \( \mathcal{E} \) is the induced EMF, and \( \Phi \) is the magnetic flux. The negative sign is due to Lenz's Law, which we'll discuss later. In our exercise, this law helps us calculate the EMF when the loop is moved out of the uniform magnetic field.
Magnetic Flux
Magnetic Flux is a measure of the strength of a magnetic field passing through a given area. It essentially tells us how much magnetic field is threaded through a surface like a loop of wire. The formula for calculating magnetic flux \( \Phi \) is:
  • \( \Phi = B \times A \times \cos(\theta) \)
Here \( B \) is the magnetic field strength, \( A \) is the area of the loop, and \( \theta \) is the angle between the magnetic field lines and the normal to the surface. In our exercise, since the magnetic field is perpendicular to the loop's plane, \( \theta \) is 0, making \( \cos(\theta) \) equal to 1. Thus, \( \Phi = B \times \pi r^2 \), where \( r \) is the radius of the loop.
Lenz's Law
Lenz's Law is crucial for determining the direction of the induced current. This law states that any induced EMF will act in a direction that opposes the change in magnetic flux that produced it. Think of it as nature's version of a pushback 鈥� when a change happens, the induced current will try to resist that change. This law complements Faraday's Law, represented by the negative sign in the formula \( \mathcal{E} = -\frac{d\Phi}{dt} \). In our problem, as the loop exits the magnetic field, the induced current will attempt to maintain the magnetic flux by generating a magnetic field in the same direction as the original.
Induced EMF
Induced EMF or electromotive force is essentially the voltage generated by changing the magnetic environment of a coil of wire. In our problem, we calculated it using Faraday's Law. By finding the change in magnetic flux and dividing by the time interval, we can find the average EMF:
  • \( \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \)
Where \( \Delta \Phi \) is the change in magnetic flux, calculated as \( \Phi_f - \Phi_i \). In this exercise, moving the loop out of the magnetic field reduced the magnetic flux to zero, resulting in a calculated induced EMF of approximately 34.05 volts.
Direction of Induced Current
The direction of induced current is a direct consequence of Lenz's Law. As we determined, the induced current opposes the change in flux resulting from moving out of the magnetic field. In practice, for a loop viewed from above, the presence and removal of an upward-directed magnetic field means that the induced current will need to reinforce the original magnetic field direction.
  • To reinforce an upward magnetic field, the induced current flows counterclockwise when observed from above.
Understanding the process of determining this direction requires visualizing the magnetic field lines and how currents interact with magnetic fields, aligning with the right-hand rule for determining direction.

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Most popular questions from this chapter

Antenna emf. A satellite, orbiting the earth at the equator at an altitude of 400 \(\mathrm{km}\) , has an antenna that can be modeled as a \(2.0-\mathrm{m}-\) long rod. The antenna is oriented perpendicular to the earth's surface. At the cquator, the earth's magnetic field is cssentially horizontal and has a value of \(8.0 \times 10^{-5} \mathrm{T}\) ; ignore any changes in \(B\) with altitude. Assuming the orbit is circular, determine the induced emf between the tips of the antenna.

It is impossible to have a uniform electric field that abruptly drops to zero in a region of space in which the magnetic field is constant and in which there are no electric charges. To prove this statement, use the method of contradiction: Assume that such a case is possible and then show that your assumption contradicts a law of nature. (a) In the bottom half of a piece of paper, draw evenly spaced horizontal lines representing a uniform electric field to your right. Use dashed lines to draw a rectangle abcda with horizontal side ab in the electric-field region and horizontal side \(c d\) in the top half of your paper where \(E=0 .\) (b) Show that integration around your rectangle contradicts Faraday's law, Eq. \((29.21) .\)

The compound \(\mathrm{SiV}_{3}\) is a type-II superconductor. At temperatures near absolute zero the two critical fields are \(B_{c 1}=\) 55.0 \(\mathrm{mT}\) and \(B_{c 2}=15.0 \mathrm{T}\) . The normal phase of \(\mathrm{SiV}_{3}\) has a magnetic susceptibility close to zero. A long, thin \(\mathrm{SiV}_{3}\) cylinder has its axis parallel to an extermal magnetic field \(\vec{B}_{0}\) 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 \(\overrightarrow{\boldsymbol{B}}\) 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?

Terminal Speed. A bar of length \(L=0.8 \mathrm{m}\) is free to shide without friction on horizontal rails, as shown in Fig. 29.48 . There is a uniform magnetic ficld \(B=1.5 \mathrm{T}\) directed into the plane of the figure. At one end of the rails there is a battery with emf \(\mathcal{E}=12 \mathrm{V}\) and a switch. The bar has mass 0.90 \(\mathrm{kg}\) and resistance \(5.0 \Omega,\) and all other resistance in the circuit can be ignored. The switch is closed at time \(t=0\) . (a) Sketch the speed of the bar as a function of time. (b) Just aner the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is 2.0 \(\mathrm{m} / \mathrm{s} ?\) (d) What is the terminal speed of the bar?

A metal ring 4.50 \(\mathrm{cm}\) in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic ficld. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 \(\mathrm{T} / \mathrm{s}\) (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?
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