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Chapter 9: Problem 2

In a physics laboratory experiment, a coil with 200 turns enclosing an area of \(12 \mathrm{~cm}^{2}\) is rotated in \(0.040 \mathrm{~s}\) 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
(a) Initial flux: 1.44脳10鈦宦� Wb; Final: 0 Wb. (b) Average emf: 0.036 V.

Step by step solution

01

Calculate Initial Magnetic Flux

The magnetic flux \( \Phi \) through the coil when it is perpendicular to the magnetic field is given by \( \Phi = NBA \), where \( N \) is the number of turns, \( B \) is the magnetic field, and \( A \) is the area.\[ \begin{align*} N &= 200 \ B &= 6.0 \times 10^{-5} \text{ T} \ A &= 12 \times 10^{-4} \text{ m}^2 \ \Phi_{\text{initial}} &= N \cdot B \cdot A = 200 \times 6.0 \times 10^{-5} \cdot 12 \times 10^{-4} \ &= 1.44 \times 10^{-3} \text{ Wb} \end{align*} \]
02

Calculate Final Magnetic Flux

When the coil is parallel to the magnetic field, the magnetic flux through the coil is zero because the angle between the area vector and the magnetic field is \( 90掳 \). Therefore:\[ \Phi_{\text{final}} = 0 \text{ Wb} \]
03

Calculate Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is the difference between the initial and final flux:\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = 0 - 1.44 \times 10^{-3} = -1.44 \times 10^{-3} \text{ Wb} \]
04

Calculate Average Induced EMF

According to Faraday's law, the average induced emf \( \mathcal{E} \) is given by the change in magnetic flux over time:\[ \mathcal{E} = - \frac{\Delta \Phi}{\Delta t} \]Given that the time \( \Delta t \) is \( 0.040 \text{ s} \), we have:\[ \mathcal{E} = - \frac{-1.44 \times 10^{-3}}{0.040} = 0.036 \text{ V} \]

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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 a fundamental concept describing the magnetic field passing through a given area. Think of it as the number of magnetic field lines crossing through a surface.
It is an important factor in calculating electromagnetic processes, like induction. The formula to calculate magnetic flux, \( \Phi \), is given by:
  • \( \Phi = B \times A \times \cos(\theta) \)
where \( B \) is the magnetic field strength, \( A \) is the area through which the field lines pass, and \( \theta \) is the angle between the field lines and the normal (perpendicular) to the surface.

In our exercise, when the coil's plane is perpendicular to the magnetic field, \( \theta = 0\), and thus \( \cos(0) = 1\) leads to maximum flux through the coil.
On rotation until the coil is parallel to the field, \( \theta = 90^\circ \), making \( \cos(90^\circ) = 0\), resulting in zero flux. Therefore, the change in orientation causes a change in the flux observed.
Induced EMF
Induced electromotive force (emf) arises when a conductor experiences a change in magnetic flux. According to Faraday's Law, the average induced emf, \( \mathcal{E} \), in a coil is directly proportional to the rate of change of the magnetic flux through the coil.

This is expressed by the formula:
  • \( \mathcal{E} = - \frac{\Delta \Phi}{\Delta t} \)
where \( \Delta \Phi \) is the change in magnetic flux and \( \Delta t \) is the time taken for this change to occur.
The negative sign indicates the direction of the induced emf, as described by Lenz's Law, which states that the induced current will oppose the change in flux.

In our scenario, as the coil rotates from perpendicular to parallel relative to the magnetic field, the flux changes, and thus an emf is induced, calculated to be \( 0.036 \text{ V} \). This is precisely the rapid shift in magnetic environment that generates electrical energy.
Coil Rotation
The coil rotation is crucial in altering the magnetic flux within the coil, leading to a variation in induced emf. When discussing coil orientation, imagine the frame within which it lies.

A coil is rotating from a perpendicular state to a parallel state relative to Earth's magnetic field. This change is key because it shifts the angle of the coil relative to the magnetic field, altering the magnetic flux:
  • Perpendicular Position: Maximum flux as magnetic lines pass directly through the coil.
  • Parallel Position: Minimum flux as lines simply "slide" past, not entering the coil.
The rotation process itself induces an emf proportional to the speed of rotation and the extent of change in flux.
A faster rotation or a more abrupt position change can result in a stronger induced emf, highlighting the dynamic nature of electromagnetism.
Earth's Magnetic Field
Earth's magnetic field acts like a giant magnet with magnetic lines that surround and enter the planet. Though quite weak compared to man-made magnets, Earth's field is essential in various electromagnetic applications.

In the context of this experiment, Earth's field provides the necessary magnetic environment for inducing emf in the coil. The strength of Earth's magnetic field at a given location is a significant factor, here noted as \( 6.0 \times 10^{-5} \text{ T} \).

This small but crucial component interacts with the coil providing the magnetic flux necessary for electromagnetic induction. Understanding Earth's magnetic field helps clarify why we observe such small-scale electromagnetic effects naturally and aids in designing sensitive experiments within lab settings.

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Most popular questions from this chapter

A flat, rectangular coil of dimensions \(l\) and \(w\) is pulled with uniform speed \(v\) through a uniform magnetic field \(B\) with the plane of its area perpendicular to the field (Fig. E9.7). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?

A coil \(4.00 \mathrm{~cm}\) in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B=(0.0120 \mathrm{~T} / \mathrm{s}) t+\left(3.00 \times 10^{-5} \mathrm{~T} / \mathrm{s}^{4}\right) t^{4}\). The coil is con- nected to a \(600-\Omega\) resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time \(t=5.00 \mathrm{~s}\) ?

Measuring Figure E9.20 Blood Flow. Blood contains positive and negative ions and thus is a conductor. A blood vessel, therefore, can be viewed as an electrical wire. We can even picture the flowing blood as a series of parallel conducting slabs whose thickness is the diameter \(d\) of the vessel moving with speed \(v\). (See Fig. E9.20.) (a) If the blood vessel is placed in a magnetic field \(B\) perpendicular to the vessel, as in the figure, show that the motional potential difference induced across it is \(\mathcal{E}=v B d .\) (b) If you expect that the blood will be flowing at \(15 \mathrm{~cm} / \mathrm{s}\) for a vessel \(5.0 \mathrm{~mm}\) in diameter, what strength of magnetic field will you need to produce a potential difference of \(1.0 \mathrm{mV} ?\) (c) Show that the volume rate of flow \((R)\) of the blood is equal to \(R=\pi \mathcal{E} d / 4 B\). (Note: Although the method developed here is useful in measuring the rate of blood flow in a vessel, it is limited to use in surgery because measurement of the potential \(\mathcal{E}\) must be made directly across the vessel.)

A square, conducting, wire loop of side \(L\), total mass \(m\), and total resistance \(R\) initially lies in the horizontal \(x y\)-plane, with corners at \((x, y, z)=(0,0,0),(0, L, 0),(L, 0,0)\), and \((L, L, 0)\). There is a uniform, upward magnetic field \(\overrightarrow{\boldsymbol{B}}=B \hat{\boldsymbol{k}}\) in the space within and around the loop. The side of the loop that extends from \((0,0,0)\) to \((L, 0,0)\) is held in place on the \(x\)-axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the net torque (magnitude and direction) that acts on the loop when it has rotated through an angle \(\phi\) from its original orientation and is rotating downward at an angular speed \(\omega .\) (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through \(90^{\circ}\) ? Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.

A long, straight solenoid with a cross-sectional area of \(8.00 \mathrm{~cm}^{2}\) is wound with 90 turns of wire per centimeter, and the windings carry a current of \(0.350 \mathrm{~A}\). A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in \(0.0400 \mathrm{~s}\). What is the average induced emf in the second winding?
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