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Chapter 30: Problem 19

An electric generator contains a coil of 100 turns of wire, each forming a rectangular loop \(50.0 \mathrm{~cm}\) by \(30.0 \mathrm{~cm}\). The coil is placed entirely in a uniform magnetic field with magnitude \(B=\) \(3.50 \mathrm{~T}\) and with \(\vec{B}\) initially perpendicular to the coil's plane. What is the maximum value of the emf produced when the coil is spun at 1000 rev/min about an axis perpendicular to \(\vec{B}\) ?

Short Answer

Expert verified
The maximum emf produced is approximately 5500 V.

Step by step solution

01

Understand the Formula for Induced EMF

The formula for the maximum electromotive force (emf) induced in a rotating coil in a magnetic field is given by:\[\text{emf}_{\text{max}} = N \cdot A \cdot B \cdot \omega\sin(\theta)\]where \(N\) is the number of turns, \(A\) is the area of the coil, \(B\) is the magnetic field, \(\omega\) is the angular velocity, and \(\theta\) is the angle, which is \(90^\circ\) for maximum emf. Therefore, \(\sin(90^\circ) = 1\). So, the formula simplifies to:\[\text{emf}_{\text{max}} = N \cdot A \cdot B \cdot \omega\]
02

Calculate the Area of the Coil

The area \(A\) of the rectangular coil is given by the product of its length and width. Given:\(\text{Length} = 50.0 \text{ cm} = 0.50 \, \text{m}\)\(\text{Width} = 30.0 \text{ cm} = 0.30 \, \text{m}\)Then,\[A = \text{Length} \times \text{Width} = 0.50 \, \text{m} \times 0.30 \, \text{m} = 0.15 \, \text{m}^2\]
03

Determine Angular Velocity

The coil rotates at 1000 revolutions per minute (rev/min). Convert this to angular velocity \(\omega\) in radians per second (rad/s):\[\omega = 1000 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}}\]\[\omega \approx 1000 \times \frac{2\pi}{60} \approx 104.72 \, \text{rad/s}\]
04

Calculate the Maximum Induced EMF

Substitute the known values into the simplified emf formula:\[\text{emf}_{\text{max}} = N \cdot A \cdot B \cdot \omega\]Given:\(N = 100\), \(A = 0.15 \, \text{m}^2\), \(B = 3.50 \, \text{T}\), \(\omega = 104.72 \, \text{rad/s}\)\[\text{emf}_{\text{max}} = 100 \times 0.15 \times 3.50 \times 104.72\approx 5500 \, \text{V}\]

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Key Concepts

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

Faraday's Law
Faraday's Law is a fundamental principle that explains how electricity can be generated using a magnetic field. According to this law, a change in magnetic flux through a coil of wire induces what is known as electromotive force (EMF) in the coil. This phenomenon forms the basis of many electrical devices such as generators and transformers.

There are some key points to remember about Faraday's Law:
  • The induced EMF is directly proportional to the rate of change of the magnetic flux.
  • The direction of the induced EMF depends on the direction of the change in the magnetic field or the direction of motion of the conductor.
  • The Law is mathematically represented as:
    \[\text{emf} = -N \frac{d\Phi_B}{dt}\]
    where \(N\) is the number of turns in the coil and \(\Phi_B\) is the magnetic flux.
For electricity to be generated efficiently, a consistent change in the magnetic field must be maintained. This can be done by rotating a coil within a magnetic field as seen in the original exercise's electric generator.
Magnetic Flux
Magnetic flux is a measure of the total magnetic field passing through a given area. It is crucial in understanding how magnetic fields interact with electrical circuits, and thus in the application of Faraday's Law for inducing EMF.

Here's what you need to know about magnetic flux:
  • It is represented by the symbol \(\Phi_B\) and is measured in Webers (Wb).
  • It is calculated as the product of the magnetic field \(B\) and the area \(A\) through which the field lines pass, and is given by:
    \[\Phi_B = B \cdot A \cdot \cos(\theta)\]
    where \(\theta\) is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
  • Maximum magnetic flux occurs when the field lines are perpendicular to the area, making \(\theta = 0^\circ\) and \(\cos(0^\circ) = 1\).
In the context of the generator problem given, the flux changes as the coil rotates, creating a continuous change in magnetic flux, which induces the EMF according to Faraday's Law.
Angular Velocity
Angular velocity is a measure of how fast something is rotating. It plays a vital role in systems where spinning motion creates an effect, like in the generation of EMF in our example.

Some important points to consider about angular velocity:
  • It is usually denoted by the symbol \(\omega\) and is measured in radians per second (rad/s).
  • It can be calculated from rotations per minute (RPM), a common measure, using the formula:
    \[\omega = \frac{2 \pi \times \text{RPM}}{60}\]
  • Higher angular velocity means a quicker rotation, which affects the rate of change of magnetic flux and thus the amplitude of the EMF.
For the solution of the original exercise, converting the coil's rotational speed from rev/min to rad/s was a crucial step in obtaining the angular velocity used in the calculation for the maximum induced EMF.

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Most popular questions from this chapter

A circular loop of wire \(10 \mathrm{~cm}\) in diameter (seen edge-on) is placed with its normal \(\vec{N}\) at an angle \(\theta=30^{\circ}\) with the direction of a uniform magnetic field \(\vec{B}\) of magnitude \(0.50\) T. The loop is then rotated such that \(\vec{N}\) rotates in a cone about the field direction at the rate 100 rev/min; angle \(\theta\) remains unchanged during the process. What is the emf induced in the loop?

Shows a closed loop of wire that consists of a pair of equal semicircles, of radius \(3.7 \mathrm{~cm}\), lying in mutually perpendicular planes. The loop was formed by folding a flat circular loop along a diameter until the two halves became perpendicular to each other. A uniform magnetic field \(\vec{B}\) of magnitude \(76 \mathrm{~m} \mathrm{~T}\) is directed perpendicular to the fold diameter and makes equal angles ( of \(45^{\circ}\) ) with the planes of the semicircles. The magnetic field is reduced to zero at a uniform rate during a time interval of \(4.5\) ms. During this interval, what are the (a) magnitude and (b) direction (clockwise or counterclockwise when viewed along the direction of \(\vec{B}\) ) of the emf induced in the loop?

At time \(t=0\), a \(45 \mathrm{~V}\) potential difference is suddenly applied to the leads of a coil with inductance \(L=50 \mathrm{mH}\) and resistance \(R=180 \Omega\). At what rate is the current through the coil increasing at \(t=1.2 \mathrm{~ms} ?\)

The inductance of a closely wound coil is such that an emf of \(3.00 \mathrm{mV}\) is induced when the current changes at the rate of \(5.00\) A/s. A steady current of \(8.00\) A produces a magnetic flux of \(40.0\) \(\mu\) Wb through each turn. (a) Calculate the inductance of the coil. (b) How many turns does the coil have?

Two straight conducting rails form a right angle. A conducting bar in contact with the rails starts at the vertex at time \(t=0\) and moves with a constant velocity of \(5.20 \mathrm{~m} / \mathrm{s}\) along them. A magnetic field with \(B=0.350 \mathrm{~T}\) is directed out of the page. Calculate (a) the flux through the triangle formed by the rails and bar at \(t=3.00 \mathrm{~s}\) and \((\mathrm{b})\) the emf around the triangle at that time. (c) If the emf is \(\mathscr{8}=a t^{n}\), where \(a\) and \(n\) are constants, what is the value of \(n ?\)
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