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Chapter 29: Problem 2

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 \(\mathrm{cm}^{2}\) is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is \(6.0 \times 10^{-5} \mathrm{T}\) . (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

Short Answer

Expert verified
7.2 x 10^-8 Tm虏 initial flux; 0 final flux; 3.6 x 10^-4 V average EMF.

Step by step solution

01

Convert Area to Square Meters

The coil's area is given in square centimeters. To use it in the magnetic flux formula, we need it in square meters. Since 1 cm虏 = 0.0001 m虏, the area is converted as follows:\[ A = 12 \, \text{cm}^2 = 12 \times 0.0001 \, \text{m}^2 = 0.0012 \, \text{m}^2 \]
02

Calculate Initial Magnetic Flux

Before the coil is rotated, its plane is perpendicular to the magnetic field, so the angle \(\theta = 0^\circ\), where \(\cos(0^\circ) = 1\). Using the formula for magnetic flux \(\Phi = B \cdot A \cdot \cos \theta\), we find:\[ \Phi_i = (6.0 \times 10^{-5} \, \text{T}) \cdot (0.0012 \, \text{m}^2) \cdot 1 = 7.2 \times 10^{-8} \, \text{T m}^2 \]
03

Calculate Final Magnetic Flux

After rotation, the coil's plane is parallel to the magnetic field, so \(\theta = 90^\circ\) and \(\cos(90^\circ) = 0\). The magnetic flux is:\[ \Phi_f = (6.0 \times 10^{-5} \, \text{T}) \cdot (0.0012 \, \text{m}^2) \cdot 0 = 0 \, \text{T m}^2 \]
04

Determine Change in Magnetic Flux

The change in magnetic flux, \(\Delta \Phi\), is given by the difference between the initial and final magnetic flux:\[ \Delta \Phi = \Phi_f - \Phi_i = 0 - 7.2 \times 10^{-8} \, \text{T m}^2 = -7.2 \times 10^{-8} \, \text{T m}^2 \]
05

Calculate Average Induced EMF

Using Faraday's Law of Induction, the average induced EMF \(\mathcal{E}\) over time \(\Delta t\) is:\[ \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \]Given \(N = 200\) turns and \(\Delta t = 0.040 \, \text{s}\), the EMF is:\[ \mathcal{E} = -200 \frac{-7.2 \times 10^{-8}}{0.040} = 3.6 \times 10^{-4} \, \text{V} \]
06

Finalize the Results

The total magnetic flux through the coil before rotation was \(7.2 \times 10^{-8} \, \text{T m}^2\), and after rotation was \(0\). The average induced EMF in the coil is \(3.6 \times 10^{-4} \, \text{V}\).

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Key Concepts

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

Magnetic Flux
Magnetic flux is an essential concept in electromagnetism and it quantifies the number of magnetic field lines passing through a defined area. Imagine it as the measure of the magnetic field's strength over a specific surface.
This is expressed mathematically as \( \Phi = B \cdot A \cdot \cos \theta \) where:
  • \( \Phi \) is the magnetic flux measured in Weber (Wb)
  • \( B \) is the magnetic field strength, in Tesla (T)
  • \( A \) is the area through which the field lines pass, in square meters (m虏)
  • \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface
In the initial position of the coil, where its plane is perpendicular to the magnetic field, \( \cos(0^\circ) = 1 \), maximizing the flux. When the coil is parallel, \( \cos(90^\circ) = 0 \), resulting in zero flux.
Thus, by rotating the coil from \( 0^\circ \) to \( 90^\circ \), the magnetic flux through it changes from its maximum value to zero.
Faraday's Law of Induction
Faraday's Law of Induction is a fundamental principle that describes how a change in magnetic flux can induce an electromotive force, or EMF, in a coil of wire. According to this law, the induced EMF in a closed loop is directly proportional to the rate of change of magnetic flux through the loop.
The formula representing Faraday's Law of Induction is given by:\[ \mathcal{E} = -N \frac{\Delta \Phi}{\Delta t} \]where:
  • \( \mathcal{E} \) is the induced EMF, in volts (V)
  • \( N \) is the number of turns in the coil
  • \( \Delta \Phi \) is the change in magnetic flux, in Weber (Wb)
  • \( \Delta t \) is the time over which the change occurs, in seconds (s)
The negative sign indicates Lenz's Law, which states that the direction of the induced EMF opposes the cause of its creation.
In our experiment where the coil changes orientation, the alteration in flux over a fraction of a second results in an induced EMF. This EMF can then create an electric current if the circuit is closed.
Electromotive Force (EMF)
Electromotive force, often abbreviated as EMF, is a key concept in the study of electromagnetism. Despite its name, EMF is not actually a force; rather, it is a potential difference or voltage developed by a source that's capable of doing electrical work when current moves between two points.
In the context of electromagnetic induction, EMF is generated due to changes in magnetic flux. When a coil of wire experiences a change in magnetic field, an EMF is induced in the coil according to Faraday's Law. This induced EMF is crucial in many electrical devices, such as generators and transformers, where it is harnessed to perform useful work.
An important thing to note is that the induced EMF depends on both the rate at which the magnetic flux changes and the number of coil turns. It is this interplay that allows for the efficient transformation of kinetic energy from rotation into electrical energy, which can then light up your room or power devices when necessary.

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Most popular questions from this chapter

A circular conducting ring with radius \(r_{0}=0.0420 \mathrm{m}\) lies in the \(x y\) -plane in a region of uniform magnetic field \(\vec{B}=B_{0}\left[1-3\left(t / t_{0}\right)^{2}+\right.\) 2\(\left(t / t_{0}\right)^{3} \hat{\boldsymbol{k}} .\) In this expression, \(t_{0}=\) 0.0100 \(\mathrm{s}\) and is constant, \(t\) is time, \(\hat{\boldsymbol{k}}\) is the unit vector in the \(+z\) - direction, and \(B_{0}=0.0800 \mathrm{T}\) and is constant. At points \(a\) and \(b(\) Fig. \(\mathrm{P} 29.64)\) there is a small gap in the ring with wires leading to an external circuit of resistance \(R=12.0 \Omega .\) There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux \(\Phi_{B}\) through the ring. (b) Determine the emf induced in the ring at time \(t=5.00 \times 10^{-3}\) s. What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through \(R\) at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time \(t=1.21 \times 10^{-2}\) s. What is the polarity of the emf? (e) Determine the time at which the current through \(R\) reverses its direction.

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 \(\mathrm{cm} .\) The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 \(\mathrm{mV}\) ?

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. E29.15. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) \(B\) is increasing; (b) \(B\) is decreasing; (c) \(B\) is constant with value \(B_{0} .\) Explain your reasoning.

A long, thin solenoid has 900 turns per meter and radius 2.50 \(\mathrm{cm} .\) The current in the solenoid is increasing at a uniform rate of 60.0 \(\mathrm{A} / \mathrm{s}\) . What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 \(\mathrm{cm}\) from the axis of the solenoid; (b) 1.00 \(\mathrm{cm}\) from the axis of the solenoid?

CALC In Fig. 29.22 the capacitor plates have area 5.00 \(\mathrm{cm}^{2}\) and separation 2.00 \(\mathrm{mm}\) . The plates are in vacuum. The charging current \(i_{\mathrm{C}}\) has a constant value of 1.80 \(\mathrm{mA} .\) At \(t=0\) the charge on the plates is zero. (a) Calculate the charge on the plates, the electric field between the plates, and the potential difference between the plates when \(t=0.500 \mu \mathrm{s}\) (b) Calculate \(d E / d t\) , the time rate of change of the electric field between the plates. Does \(d E / d t\) vary in time? (c) Calculate the displacement current density \(j_{\mathrm{D}}\) between the plates, and from this the total displacement current \(i_{\mathrm{D}} .\) How do \(i_{\mathrm{C}}\) and \(i_{\mathrm{D}}\) compare?
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