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Chapter 29: Problem 25

Are Motional emfs a Practical Source of Electricity? How fast (in \(\mathrm{m} / \mathrm{s}\) and mph) would a 5.00 -cm copper bar have to move at right angles to a \(0.650-\mathrm{T}\) magnetic field to generate 1.50 \(\mathrm{V}\) (the same as a A \(\mathrm{A}\) battery) across its ends? Does this seem like a practical way to generate electricity?

Short Answer

Expert verified
The bar needs to move at 46.15 m/s or about 103.25 mph. This is impractical for generating electricity.

Step by step solution

01

Understand the Problem

We need to determine how fast a 5.00 cm copper bar needs to move perpendicular to a magnetic field of 0.650 T to generate 1.50 V. We are asked if this is a practical way to generate electricity.
02

Formula for Motional EMF

Motional EMF can be calculated using the formula: \( \varepsilon = B \cdot l \cdot v \), where \( \varepsilon \) is the EMF, \( B \) is the magnetic field strength, \( l \) is the length of the bar, and \( v \) is the speed of the bar. We need to solve for \( v \).
03

Rearrange the Formula

Rearrange the formula to solve for \( v \): \( v = \frac{\varepsilon}{B \cdot l} \).
04

Substitute Values and Solve for \( v \)

Substitute the given values into the equation: \( \varepsilon = 1.50 \) V, \( B = 0.650 \) T, and \( l = 0.05 \) m. Calculate: \[ v = \frac{1.50}{0.650 \times 0.05} = 46.15 \text{ m/s} \].
05

Convert m/s to mph

Convert the speed from meters per second to miles per hour using the conversion: 1 m/s \( \approx 2.237 \) mph. Thus, \( 46.15 \text{ m/s} \times 2.237 \approx 103.25 \text{ mph} \).
06

Assess Practicality

Considering that the bar needs to move at 103.25 mph to generate only 1.50 V, this is practically inefficient and unrealistic for generating significant electricity.

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Key Concepts

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

Electromagnetic Induction
Electromagnetic induction is a fascinating concept, central to understanding motional EMF (electromotive force). It refers to the generation of voltage across a conductor when it is exposed to a changing magnetic field. This principle was first discovered by Michael Faraday and is governed by Faraday's law of induction.
When a conductor, like a copper bar, moves through a magnetic field perpendicular to its direction, it cuts through magnetic lines of force. As a result, an EMF is induced in the conductor. The direction of the induced EMF is given by Lenz's Law, which states that the induced EMF will always oppose the change in magnetic flux.
In practical terms, electromagnetic induction is the principle behind transformers, inductors, and many types of electrical generators. However, when applied to motional EMF, the practicality often depends on the speed and conditions required to generate a usable voltage. This is crucial in understanding why motional emfs, in some scenarios like the one involving a moving copper bar, might not be practical despite the interesting theoretical foundation of electromagnetic induction itself.
Magnetic Fields
Magnetic fields are invisible forces that exert a force on moving charges, like electrons in a conductor. A magnetic field is characterized by its strength, measured in Tesla (T), and its direction. This field is visualized as lines that go from the north to the south pole of a magnet.
When you place a conductor like our copper bar inside a magnetic field, and if you move it perpendicularly, it disturbs these magnetic field lines, causing electrons within the conductor to experience a force. This force is what induces a voltage or EMF across the conductor鈥檚 ends. The stronger the magnetic field, the greater potential for generating a larger EMF.
The principle is used in many technologies, such as MRI machines and electric motors, where control over magnetic fields allows precise manipulation of magnetic force. In the exercise, the magnetic field strength is 0.650 T, and the movement of the copper bar perpendicular to it is essential for inducing the specified voltage.
Electric Potential
Electric potential is the potential energy per unit charge due to the position of a charge in an electric field. It is crucial for understanding how energy is stored and transferred in electrical circuits.
The electric potential difference between two points is what drives current through a circuit. It is measured in volts (V) and is often referred to as 'voltage'. The EMF developed when a conductor moves through a magnetic field is a type of electric potential difference.
In the scenario of the copper bar, when it moves through the magnetic field, an EMF of 1.50 V could be generated. This EMF is a result of the work done on the charge carriers by the magnetic field. However, achieving this small voltage requires significant speed, illustrating the challenge of using motional EMF as a practical source of electric potential in most everyday applications.
Understanding electric potential helps us appreciate how devices like batteries store energy and how generated differences, such as those created by electromagnetic induction, lead to usable electrical energy.

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Most popular questions from this chapter

CALC 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 connected 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 \(t=5.00 \mathrm{s?}\)

At temperatures near absolute zero, \(B_{\mathrm{c}}\) approaches 0.142 \(\mathrm{T}\) for vanadium, a type-I superconductor. The normal phase of vanadium has a magnetic susceptibility close to zero. Consider a long, thin vanadium cylinder with its axis parallel to an external magnetic field \(\vec{B}_{0}\) in the \(+x\) -direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the \(x\) -axis. At temperatures near absolute zero, what are the resultant magnetic field \(\vec{B}\) and the magnetization \(\vec{M}\) inside and outside the cylinder (far from the ends ) for (a) \(\vec{B}_{0}=(0.130 \mathrm{T}) \hat{\imath}\) and (b) \(\vec{\boldsymbol{B}}_{0}=(0.260 \mathrm{T}) \hat{\mathfrak{t}} ?\)

CALC In a region of space, a magnetic field points in the \(+x\) -direction (toward the right). Its magnitude varies with position according to the formula \(B_{x}=B_{0}+b x,\) where \(B_{0}\) and \(b\) are positive constants, for \(x \geq 0 .\) A flat coil of area \(A\) moves with uniform speed \(v\) from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

A long, thin solenoid has 400 turns per meter and radius 1.10 \(\mathrm{cm} .\) The current in the solenoid is increasing at a uniform rate \(d i / d t .\) The induced electric field at a point near the center of the solenoid and 3.50 \(\mathrm{cm}\) from its axis is \(8.00 \times 10^{-6} \mathrm{V} / \mathrm{m}\) . Calculate di/dt.

Search Coils and Credit Cards. One practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. The coil is initially held with its plane perpendicular to a magnetic field. The coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) Derive the equation relating the total charge \(Q\) that flows through a search coil to the magnetic-field magnitude \(B\) . The search coil has \(N\) turns, each with area \(A,\) and the flux through the coil is decreased from its initial maximum value to zero in a time \(\Delta t .\) The resistance of the coil is \(R,\) and the total charge is \(Q=I \Delta t,\) where \(I\) is the average current induced by the change in flux. (b) In a credit card reader, the magnetic strip on the back of a credit card is rapidly "swiped" past a coil within the reader. Explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) Is it necessary that the credit card be "swiped" through the reader at exactly the right speed? Why or why not?
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