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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

A \(120.0-\mathrm{V}\) motor draws a current of 7.00 A when running at normal speed. The resistance of the armature wire is \(0.720 \Omega\). (a) Determine the back emf generated by the motor. (b) What is the current at the instant when the motor is just turned on and has not begun to rotate? (c) What series resistance must be added to limit the starting current to \(15.0 \mathrm{~A} ?\)

a constant current \(I\) exists in a solenoid whose inductance is \(L\). The current is then reduced to zero in a certain amount of time. (a) If the wire from which the solenoid is made has no resistance, is there a voltage across the solenoid during the time when the current is constant? (b) If the wire from which the solenoid is made has no resistance, is there an emf across the solenoid during the time that the current is being reduced to zero? (c) Does the solenoid store electrical energy when the current is constant? If so, express this energy in terms of the current and the inductance. (d) When the current is reduced from its constant value to zero, what is the rate at which energy is removed from the solenoid? Express your answer in terms of the initial current, the inductance, and the time during which the current goes to zero. A solenoid has an inductance of \(L=3.1 \mathrm{H}\) and carries a current of \(I=15 \mathrm{~A}\). (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 zero in \(75 \mathrm{~ms} ?\)

The drawing shows a coil of copper wire that consists of two semicircles joined by straight sections of wire. In part \(a\) the coil is lying flat on a horizontal surface. The dashed line also lies in the plane of the horizontal surface. Starting from the orientation in part \(a\), the smaller semicircle rotates at an angular frequency \(\omega\) about the dashed line, until its plane becomes perpendicular to the horizontal surface, as shown in part \(b\). A uniform magnetic field is constant in time and is directed upward, perpendicular to the horizontal surface. The field completely fills the region occupied by the coil in either part of the drawing. (a) In which part of the drawing, if either, does a greater magnetic flux pass through the coil? Account for your answer. (b) As the shape of the coil changes from that in part \(a\) of the drawing to that in part \(b\), does an induced current flow in the coil, and, if so, in which direction does it flow? Give your reasoning. To describe the flow, imagine that you are above the coil looking down at it. (c) How is the period \(T\) of the rotational motion related to the angular frequency \(\omega\), and in terms of the period, what is the shortest time interval that elapses between parts \(a\) and \(b\) of the drawing? the magnitude of the magnetic field is \(0.35 \mathrm{~T}\). The resistance of the coil is \(0.025 \Omega\), and the smaller semicircle has a radius of \(0.20 \mathrm{~m}\). The angular frequency at which the small semicircle rotates is \(1.5 \mathrm{rad} / \mathrm{s}\). Determine the average current, if any, induced in the coil as the coil changes shape from that in part \(a\) of the drawing to that in \(\operatorname{part} b\).

Parts \(a\) and \(b\) of the drawing show the same uniform and constant (in time) magnetic field \(\overrightarrow{\mathbf{B}}\) directed perpendicularly into the paper over a rectangular region. Outside this region, there is no field. Also shown is a rectangular coil (one turn), which lies in the plane of the paper. In part \(a\) the long side of the coil (length \(=L)\) is just at the edge of the field region, while in part \(b\) the short side (width \(=W\) is just at the edge. It is known that \(L / W=3.0 .\) In both parts of the drawing the coil is pushed into the field with the same velocity \(\overrightarrow{\mathbf{v}}\) until it is completely within the field region. The magnitude of the average emf induced in the coil in part \(a\) is \(0.15 \mathrm{~V}\). What is its magnitude in part \(b\) ?

A motor is designed to operate on \(117 \mathrm{~V}\) and draws a current of \(12.2 \mathrm{~A}\) when it first starts up. At its normal operating speed, the motor draws a current of \(2.30\) A. Obtain (a) the resistance of the armature coil, (b) the back emf developed at normal speed, and (c) the current drawn by the motor at one-third normal speed.
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