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Chapter 25: Problem 10

A closely wound rectangular coil of 80 turns has dimensions of \(25.0 \mathrm{~cm}\) by \(40.0 \mathrm{~cm} .\) The plane of the coil is rotated from a position where it makes an angle of \(37.0^{\circ}\) with a magnetic field of \(1.70 \mathrm{~T}\) to a position perpendicular to the field. The rotation takes \(0.0600 \mathrm{~s}\). What is the average emf induced in the coil?

Short Answer

Expert verified
The average emf induced in the coil is found using Faraday's law. After deriving the change in magnetic flux, the absolute value of the average emf can be calculated as: \(\epsilon = N \cdot \Delta \Phi / \Delta t\). The desired short answer must be calculated using the results from the step 3.

Step by step solution

01

Derive the magnetic flux before rotation

First, we need to calculate the magnetic flux \(\Phi\) before the rotation. The formula for magnetic flux is \(\Phi = B \cdot A \cdot \cos(\theta)\), where \(B\) is the magnetic field, \(A\) is the area of the coil, and \(\theta\) is the angle between the magnetic field and the normal to the coil. In this case, \(B = 1.70 T\), \(\theta = 37.0掳\), and \(A = 25.0 cm \times 40.0 cm = 0.1 m^2\). Substituting these values gives: \(\Phi_{before} = B \cdot A \cdot \cos(\theta) = 1.70 T \cdot 0.1 m^2 \cdot \cos(37.0掳)\)
02

Derive the magnetic flux after rotation

Next, we determine the magnetic flux after the rotation. Because the coil is rotated to a position perpendicular to the magnetic field, the angle \(\theta = 0^{\circ}\). Substituting the values into the same equation as before, we get: \(\Phi_{after} = B \cdot A \cdot \cos(\theta) = 1.70 T \cdot 0.1 m^2 \cdot \cos(0掳)\)
03

Calculate the change in magnetic flux

The change in magnetic flux \(\Delta \Phi\) is given by: \(\Delta \Phi = \Phi_{after} - \Phi_{before}\). Substituting the results from step 1 and step 2 into this equation gives the change in magnetic flux.
04

Derive the average emf

Finally, we can calculate the average emf using Faraday's law. The law states that the induced emf equals the change in magnetic flux over the change in time, or \(\epsilon = -N \cdot \Delta \Phi / \Delta t\), where \(N\) is the number of turns in the coil and \(\Delta t\) is the time it takes for the flux change to occur. In this case, \(N = 80\) turns and \(\Delta t = 0.0600 s\). Substituting the result from step 3 into this equation gives the average emf. Note, the negative sign in the formula indicates that the emf creates a current \(I\) that opposes the change in \(\Phi\) (as per Lenz's law), but as we are only asked for the magnitude of the emf, we can ignore the sign.

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Key Concepts

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

Faraday's Law
One of the fundamental principles of electromagnetic induction is Faraday's Law. This law elegantly relates the rate of change of magnetic flux through a coil to the induced electromotive force (emf) in the coil.
The formula for Faraday's Law is expressed as \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \), where:
  • \( \varepsilon \): Induced emf
  • \(N\): The number of turns in the coil
  • \( \Delta \Phi \): The change in magnetic flux
  • \( \Delta t \): The time duration of the change
The negative sign in the formula indicates Lenz's Law, which suggests that the direction of the induced current will oppose the change in the magnetic flux. However, when finding just the magnitude of the emf, this sign is often ignored in calculations.
Faraday's Law provides a quantitative relationship allowing us to predict how effective a coil will be in converting the energy of a changing magnetic field into electrical energy.
Magnetic Flux
Magnetic flux is a concept that measures the total magnetic field passing through a given area. It is an essential component in understanding electromagnetic induction, as it directly influences the induced emf.
The formula to calculate magnetic flux \( \Phi \) is: \( \Phi = B \times A \times \cos(\theta) \), where:
  • \(B\): The magnetic field strength
  • \(A\): The area of the coil
  • \(\theta\): The angle between the field and the normal to the coil
By varying the angle or the field strength, we can affect the total magnetic flux through a coil.
In problems involving electromagnetic induction, such as calculating the induced emf in a rotating coil, the change in magnetic flux as the coil changes its position relative to the magnetic field is critical.
Understanding magnetic flux allows us to analyze how mechanical motion (like rotation) can influence the electrical properties of a system.
Lenz's Law
A key aspect of Faraday's Law is its connection to Lenz's Law, which emphasizes the direction of the induced emf. Lenz's Law is a conservation rule ensuring that energy is preserved in electromagnetic systems.
According to Lenz's Law, the direction of the induced current (and thus the induced emf) will always act to oppose the change that produced it. This behavior is why we include a negative sign in Faraday's formula: \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \).
Lenz's Law can be thought of as nature's way of maintaining balance and preventing "free energy" from being created out of nowhere.
In practical applications, such as electric generators and motors, Lenz's Law plays a crucial role in determining how designs can efficiently manage energy transfer through electromagnetic induction.

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Most popular questions from this chapter

A very long, straight solenoid with a cross-sectional area of \(2.00 \mathrm{~cm}^{2}\) is wound with 90.0 turns of wire per centimeter. Starting at \(t=0\) the current in the solenoid is increasing according to \(i(t)=\left(0.160 \mathrm{~A} / \mathrm{s}^{2}\right) t^{2}\). A secondary winding of 5 turns encircles the solenoid at its center, such that the secondary winding has the same cross-sectional area as the solenoid. What is the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is \(3.20 \mathrm{~A}\) ?

The battery for a certain cell phone is rated at \(3.70 \mathrm{~V}\). According to the manufacturer it can produce \(3.15 \times 10^{4} \mathrm{~J}\) of electrical energy, enough for \(5.25 \mathrm{~h}\) of operation, before needing to be recharged. Find the average current that this cell phone draws when turned on.

A battery-powered global positioning system (GPS) receiver operating on \(9.0 \mathrm{~V}\) draws a current of 0.13 A. How much electrical energy does it consume during 30 minutes?

Light Bulbs. The power rating of a light bulb (such as a \(100 \mathrm{~W}\) bulb is the power it dissipates when connected across a \(120 \mathrm{~V}\) potential difference. What is the resistance of (a) a \(100 \mathrm{~W}\) bulb and (b) a \(60 \mathrm{~W}\) bulb? (c) How much current does each bulb draw in normal use?

In another experiment, a piece of the web is suspended so that it can move freely. When either a positively charged object or a negatively charged object is brought near the web, the thread is observed to move toward the charged object. What is the best interpretation of this observation? The web is (a) a negatively charged conductor; (b) a positively charged conductor; (c) either a positively or negatively charged conductor; (d) an electrically neutral conductor.
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