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Chapter 22: Problem 23

mmh A constant magnetic field passes through a single rectangular loop whose dimensions are 0.35 \(\mathrm{m} \times 0.55 \mathrm{m}\) . The magnetic field has a magnitude of 2.1 \(\mathrm{T}\) and is inclined at an angle of \(65^{\circ}\) with respect to the normal to the plane of the loop. (a) If the magnetic field decreases to zero in a time of 0.45 \(\mathrm{s}\) , what is the magnitude of the average emf induced in the loop? (b) If the magnetic field remains constant at its initial value of 2.1 \(\mathrm{T}\) , what is the magnitude of the rate \(\Delta A / \Delta t\) which the area should change so that the average emf has the same magnitude as in part (a)?

Short Answer

Expert verified
The average emf induced is approximately 0.45 V, and the area must change at a rate of 2.62 m虏/s to maintain the same emf.

Step by step solution

01

Understanding the Problem: Part (a)

To find the magnitude of the average induced emf when the magnetic field decreases to zero, we apply Faraday's law of electromagnetic induction. First, calculate the initial magnetic flux, then determine the change in flux and divide it by the time interval.
02

Calculating Initial Magnetic Flux

The initial magnetic flux is given by \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B = 2.1 \, \text{T} \), \( A = 0.35 \, \text{m} \times 0.55 \, \text{m} \), and \( \theta = 65^{\circ} \). Calculate the area: \( A = 0.35 \times 0.55 = 0.1925 \, \text{m}^2 \). Then, \( \Phi = 2.1 \times 0.1925 \times \cos(65^{\circ}) \).
03

Calculating Flux Change

The magnetic flux goes from an initial value \( \Phi_{initial} \) to zero. Thus, \( \Delta \Phi = 0 - \Phi_{initial} \).
04

Applying Faraday's Law for Induced EMF

Use Faraday's law: \( \mathcal{E}_{avg} = - \frac{\Delta \Phi}{\Delta t} \). Since magnitude is being asked, \( \mathcal{E}_{avg} = \frac{\Phi_{initial}}{0.45 \, \text{s}} \). Substitute the values to find the average emf.
05

Understanding Part (b)

With the magnetic field constant at 2.1 T, determine the rate of change of area \( \frac{\Delta A}{\Delta t} \) that yields the same average emf calculated in Part (a).
06

Relating EMF and Rate of Area Change

Since \( \mathcal{E}_{avg} = B \cdot \frac{\Delta A}{\Delta t} \cdot \cos(\theta) \), rearrange to find \( \frac{\Delta A}{\Delta t} = \frac{\mathcal{E}_{avg}}{B \cdot \cos(\theta)} \). Substitute the known values to compute \( \frac{\Delta A}{\Delta t} \).

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Key Concepts

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

Magnetic Flux
Magnetic flux is a measure of the amount of magnetic field that passes through a given area. It is a key concept in understanding electromagnetic induction. The magnetic flux \( \Phi \) through a surface can be defined as:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
  • \( B \) is the magnetic field strength.
  • \( A \) is the area through which the field lines pass.
  • \( \theta \) is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
In the exercise, you begin by calculating the initial magnetic flux through the rectangular loop. This is essential to finding the induced EMF as changes in magnetic flux drive this process as per Faraday's Law.
Induced EMF
Induced electromotive force (EMF) is the voltage generated by changing magnetic flux in a closed loop. According to Faraday's Law of Electromagnetic Induction, a change in magnetic flux through a loop induces an EMF. The law states:
\[ \mathcal{E} = - \frac{d\Phi}{dt} \]
  • \( \mathcal{E} \) is the induced EMF.
  • \( d\Phi \) is the change in magnetic flux.
  • \( dt \) is the change in time over which the flux changes.
In this problem, you first calculate the initial flux and then find its change over the given time to compute the average induced EMF. It's noteworthy how the direction of an induced EMF (given by Lenz鈥檚 Law) opposes the change causing it, though here we're only concerned with the magnitude.
Rate of Change of Area
The rate of change of area is vital when the magnetic field remains constant but we wish to maintain the induced EMF equivalent to that obtained with a varying magnetic field. The change in the area that a loop covers can similarly affect magnetic flux, just like changing the field itself:
To maintain constant flux variation, Faraday鈥檚 Law adapts to:
  • \[ \mathcal{E}_{avg} = B \cdot \frac{\Delta A}{\Delta t} \cdot \cos(\theta) \]
Here, solving for \( \Delta A/\Delta t \) shows how the area change leads to a consistent induced EMF magnitude, particularly relevant when magnetic flux through a constant magnetic field needs alteration to simulate decreasing field effects.
Magnetic Field
A magnetic field (\( B \)) is an invisible force field surrounding magnetic objects, exerting forces on moving electric charges and affecting their trajectories. It is measured in Teslas (T). Understanding how this field interacts with loops of wire is crucial in electromagnetism.
In this exercise, the magnetic field was initially at \( 2.1 \, \text{T} \), tilting the calculations for magnetic flux and induced EMF:
  • The angle of \( 65^{\circ} \) between the field and the normal to the loop is critical in determining the effective component of the field aiding in flux calculation.
  • How the field's reduction to zero leads you to consider mechanisms for maintaining induced EMF even with a constant field.
These elements highlight the interplay between a magnetic field鈥檚 direction/magnitude and the conductor's alignment, integral in electromagnetic systems.

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Most popular questions from this chapter

A flat coil of wire has an area \(A, N\) turns, and a resistance \(R .\) It is situated in a magnetic field, such that the normal to the coil is parallel to the magnetic field. The coil is then rotated through an angle of \(90^{\circ},\) so that the normal becomes perpendicular to the magnetic field. The coil has an area of \(1.5 \times 10^{-3} \mathrm{m}^{2}, 50\) turns, and a resistance of 140\(\Omega .\) During the time while it is rotating, a charge of \(8.5 \times 10^{-5}\) C flows in the coil. What is the magnitude of the magnetic field?

ssm The earth's magnetic field, like any magnetic field, stores energy. The maximum strength of the earth's field is about \(7.0 \times 10^{-5} \mathrm{T}\) . Find the maximum magnetic energy stored in the space above a city if the space occupies an area of \(5.0 \times 10^{8} \mathrm{m}^{2}\) and has a height of 1500 \(\mathrm{m}\) .

A constant current of \(I=15\) A exists in a solenoid whose inductance is \(L=3.1 \mathrm{H}\) . The current is then reduced to zero in a certain amount of time. \((\mathrm{a})\) If the current goes from 15 to 0 \(\mathrm{A}\) in a time of \(75 \mathrm{ms},\) what is the emf induced in the solenoid? (b) How much electrical energy is stored in the solenoid? (c) At what rate must the electrical energy be removed from the solenoid when the current is reduced to 0 \(\mathrm{A}\) in a time of 75 \(\mathrm{ms}\) ? Note that the rate at which energy is removed is the power.

A flat circular coil with 105 turns, a radius of \(4.00 \times 10^{-2} \mathrm{m}\) and a resistance of 0.480\(\Omega\) is exposed to an external magnetic field that is directed perpendicular to the plane of the coil. The magnitude of the external magnetic field is changing at a rate of \(\Delta B / \Delta t=0.783 \mathrm{T} / \mathrm{s}\) , thereby inducing a current in the coil. Find the magnitude of the magnetic field at the center of the coil that is produced by the induced current.

The rechargeable batteries for a laptop computer need a much smaller voltage than what a wall socket provides. Therefore, a transformer is plugged into the wall socket and produces the necessary voltage for charging the batteries. The batteries are rated at \(9.0 \mathrm{V},\) and a current of 225 \(\mathrm{mA}\) is used to charge them. The wall socket provides a voltage of 120 \(\mathrm{V}\) (a) Determine the turns ratio of transformer. (b) What is the current coming from the wall socket? (c) Find the average power delivered by the wall socket and the average power sent to the batteries.
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