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Chapter 22: Problem 18

A rectangular loop of wire with sides 0.20 and \(0.35 \mathrm{~m}\) lies in a plane perpendicular to a constant magnetic field (see part \(a\) of the drawing). The magnetic field has a magnitude of \(0.65 \mathrm{~T}\) and is directed parallel to the normal of the loop's surface. In a time of \(0.18 \mathrm{~s}\), one-half of the loop is then folded back onto the other half, as indicated in part \(b\) of the drawing. Determine the magnitude of the average emf induced in the loop.

Short Answer

Expert verified
The average induced emf is 0.126 V.

Step by step solution

01

Understand the Problem

We have a rectangular loop of wire with dimensions 0.20 m and 0.35 m placed in a perpendicular magnetic field of 0.65 T. The loop is folded in half, reducing its effective area, and we need to find the average emf induced during this process.
02

Recall Faraday's Law of Induction

Faraday's Law states that the induced emf (\( \varepsilon \)) in a closed circuit is equal to the negative change in magnetic flux (\( \Phi \)) through the circuit over time. It can be expressed as:\[ \varepsilon = -\Delta \Phi / \Delta t \]
03

Calculate Initial Magnetic Flux

The initial flux (\( \Phi_i \)) is given by the product of the magnetic field (\( B \)) and the initial area (\( A_i \)) of the loop. The area is 0.20 m \( \times \) 0.35 m.\[ \Phi_i = B \times A_i = 0.65 \times (0.20 \times 0.35) = 0.65 \times 0.07 = 0.0455 \text{ Wb (Weber)} \]
04

Calculate Final Magnetic Flux

After folding, the effective area of the loop becomes half of the original area, so the final area (\( A_f \)) is 0.20 m \( \times \) 0.175 m.\[ \Phi_f = B \times A_f = 0.65 \times 0.20 \times 0.175 = 0.65 \times 0.035 = 0.02275 \text{ Wb} \]
05

Compute Change in Magnetic Flux

Calculate the change in flux (\( \Delta \Phi \)) by finding the difference between the initial and final flux.\[ \Delta \Phi = \Phi_f - \Phi_i = 0.02275 - 0.0455 = -0.02275 \text{ Wb} \]
06

Calculate Average EMF

Using Faraday's Law, compute the average emf using the change in flux and the time interval (\( \Delta t \) = 0.18 s).\[ \varepsilon = - \Delta \Phi / \Delta t = -(-0.02275) / 0.18 = 0.02275 / 0.18 \approx 0.126 \text{ V} \]
07

Finalize Solution

The magnitude of the average induced emf in the loop during the folding process is approximately 0.126 V.

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Key Concepts

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

Magnetic Flux
Magnetic flux helps us understand how much of a magnetic field passes through an area, such as a loop of wire. Imagine a magnetic field as a collection of invisible lines passing around us. The magnetic flux quantifies how many of these lines penetrate a given area.
The formula for magnetic flux \( \Phi \) is given by:
  • \( \Phi = B \times A \times \cos(\theta) \)
  • Where \( B \) is the magnetic field strength in teslas (T), \( A \) is the area in square meters (m\(^2\)), and \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the surface.
In the given scenario, the loop is initially aligned such that the magnetic field is perpendicular to it, making \( \theta = 0 \) degrees. This means \( \cos(0) = 1 \), so the initial magnetic flux directly depends on the product of \( B \) and \( A \).
This concept is at the heart of understanding electromagnetic induction and plays a crucial role in Faraday's Law of Induction.
Electromotive Force (EMF)
Electromotive force (emf) is a measure of the energy provided by a source of electric power per unit charge. Despite its name, it's not actually a force; rather, it is the potential difference that causes current to flow in a circuit.
Within the context of electromagnetic induction, emf is generated when there is a change in magnetic flux through a conductor. Faraday's Law of Induction gives us the relationship:
  • \( \varepsilon = -\Delta \Phi / \Delta t \)
This equation shows that the negative rate of change of magnetic flux, \( \Delta \Phi \), over the time interval \( \Delta t \) leads to an induced emf, \( \varepsilon \). This change can be due to the conductor moving through the magnetic field, or the magnetic field strength changing, or changes in the area of the loop (as is the case in this exercise).
The negative sign in Faraday鈥檚 Law reflects Lenz's Law, indicating that the induced emf will create a current whose magnetic field opposes the change in flux that produced it, essentially trying to keep the magnetic environment constant.
Rectangular Loop of Wire
A rectangular loop of wire is a simple and common setup in physics to study electromagnetic phenomena. Its geometric shape makes mathematical computations straightforward, particularly when considering areas and orientations concerning the magnetic field.
In this problem, the loop has sides of 0.20 m and 0.35 m, lying initially in a plane perpendicular to the magnetic field of 0.65 T. When it is folded, it affects the effective area through which the magnetic field passes.
  • Initial Area \( A_i = 0.20 \times 0.35 = 0.070 \text{ m}^2 \)
  • After folding, new effective area \( A_f = 0.20 \times 0.175 = 0.035 \text{ m}^2 \)
This change in area directly influences the magnetic flux, which subsequently affects the induced emf as described earlier. Understanding how the loop's geometry interacts with the magnetic field provides insights into the physics of changing magnetic environments.

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Most popular questions from this chapter

Interactive Solution \(\underline{22.55}\) at offers one approach to problems such as this one. The secondary coil of a step-up transformer provides the voltage that operates an electrostatic air filter. The turns ratio of the transformer is 50: 1 . The primary coil is plugged into a standard \(120-\mathrm{V}\) outlet. The current in the secondary coil is \(1.7 \times 10^{-3} \mathrm{~A} .\) Find the power consumed by the air filter.

The armature of an electric drill motor has a resistance of \(15.0 \Omega\). When connected to a 120.0-V outlet, the motor rotates at its normal speed and develops a back emf of \(108 \mathrm{~V}\). (a) What is the current through the motor? (b) If the armature freezes up due to a lack of lubrication in the bearings and can no longer rotate, what is the current in the stationary armature? (c) What is the current when the motor runs at only half speed?

Concept Questions The drawing shows a straight wire carrying a current \(I\). Above the wire is a rectangular loop that contains a resistor \(R\). (a) Does the magnetic field produced by the current \(I\) penetrate the loop and generate a magnetic flux? (b) When is there an induced current in the loop, if the current \(I\) is constant or if it is decreasing in time? (c) When there is an induced magnetic field produced by the loop, does it always have a direction that is opposite to the direction of the magnetic field produced by the current \(I\) ? Provide a reason for each answer. Problem If the current \(I\) is decreasing in time, what is the direction of the induced current through the resistor \(R\) - left to right or right to left? Give your reasoning.

A magnetic field is passing through a loop of wire whose area is \(0.018 \mathrm{~m}^{2}\). The direction of the magnetic field is parallel to the normal to the loop, and the magnitude of the field is increasing at the rate of \(0.20 \mathrm{~T} / \mathrm{s}\). (a) Determine the magnitude of the emf induced in the loop. (b) Suppose the area of the loop can be enlarged or shrunk. If the magnetic field is increasing as in part (a), at what rate (in \(\mathrm{m}^{2} / \mathrm{s}\) ) should the area be changed at the instant when \(B=1.8 \mathrm{~T}\) if the induced emf is to be zero? Explain whether the area is to be enlarged or shrunk.

The coil within an ac generator has an area per turn of \(1.2 \times 10^{-2} \mathrm{~m}^{2}\) and consists of 500 turns. The coil is situated in a 0.13-T magnetic field and is rotating at an angular speed of \(34 \mathrm{rad} / \mathrm{s}\). What is the emf induced in the coil at the instant when the normal to the loop makes an angle of \(27^{\circ}\) with respect to the direction of the magnetic field?
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