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Chapter 22: Problem 2

Near San Francisco, where the vertically downward component of the earth's magnetic field is \(4.8 \times 10^{-5} \mathrm{T}\) , a car is traveling forward at 25 \(\mathrm{m} / \mathrm{s}\) . The width of the car is 2.0 \(\mathrm{m}\) . (a) Find the emf induced between the two sides of the car. \((\mathrm{b})\) Which side of the car is positive-the driver's side or the passenger's side?

Short Answer

Expert verified
(a) 2.4 mV is the induced EMF. (b) The passenger's side is positive.

Step by step solution

01

Understanding the Problem

We need to compute the electromotive force (EMF) induced between the two sides of a car moving through a magnetic field. The magnetic field is vertical, the car's motion is horizontal, and we use the relationship between motion, magnetic fields, and induced EMF.
02

Using EMF Formula

The formula for the induced EMF in a moving conductor in a magnetic field is given by \[\text{EMF} = B \cdot v \cdot L\]where \( B = 4.8 \times 10^{-5} \mathrm{T} \) (magnetic field strength), \( v = 25 \mathrm{m/s} \) (speed of the car), and \( L = 2.0 \mathrm{m} \) (width of the car).
03

Plug in the Values

Substitute the given values into the formula:\[\text{EMF} = (4.8 \times 10^{-5} \mathrm{T}) \times (25 \mathrm{m/s}) \times (2.0 \mathrm{m})\]Calculate the result to find the EMF.
04

Calculating the EMF

Carrying out the calculation:\[\text{EMF} = 2.4 \times 10^{-3} \mathrm{V} = 2.4 \mathrm{mV}\]Thus, the EMF induced between the sides of the car is 2.4 mV.
05

Determine the Positive Side

Using the right-hand rule, we can determine the direction of the induced current. Point your fingers in the direction of velocity (forward) and curl them toward the direction of the magnetic field (downward). Your thumb points to the side with the positive charge.
06

Apply the Right-Hand Rule

When applying the right-hand rule as described, your thumb will point toward the passenger's side. Therefore, the passenger's side is positive.

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Key Concepts

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

Magnetic Field
Magnetic fields are invisible lines of force surrounding magnets, electrical currents, and moving electric charges. These fields are created by moving charges and affect other nearby charges. Understandably, magnetic fields have both a direction and a magnitude.
The magnitude, or strength, of a magnetic field, is measured in Tesla (T). In our example near San Francisco, the earth's vertical magnetic field is given as 4.8 x 10^{-5} T. This value indicates how strong the field is at that location. It's crucial to consider the direction as well, which, in the given problem, is downward.
Magnetic fields can interact with conductive materials. When a conductive material moves through a magnetic field, it experiences a change in the magnetic environment around it, which can lead to other electrical effects like induced electromotive force (EMF).
  • The field affects any moving charge or conductor.
  • Influences how induced currents and voltages develop in conductors.
Right-Hand Rule
The right-hand rule is a simple tool for determining the direction of forces in a magnetic field. It's a mental trick often used in physics when working with electricity and magnetism. In essence, it helps to predict how and where various interactions happen.
Here's how you apply it: stretch out your right hand. Point your fingers in the direction of the velocity of the moving object, such as a car. Then, curl your fingers toward the direction of the magnetic field. Your thumb, sticking out at a right angle, then points in the direction of the induced current or the side where the positive charge will appear.
  • It helps visualize the direction of the current.
  • Useful for figuring out directions without complex calculations.
This simple yet powerful technique is incredibly handy in physics, especially when trying to determine the effects of magnetic forces as in the given problem.
Horizontal Motion
Horizontal motion in physics refers to movement along a straight line parallel to the horizon. In our scenario, the car is traveling horizontally with a constant speed of 25 m/s.
This motion, combined with a magnetic field, affects the charges inside the vehicle. As the car travels forward, each charge inside experiences a magnetic force. Since the car is entirely in motion through the magnetic field, this movement induces an electromotive force across its width.
In simple terms, horizontal motion in a magnetic field leads to separation of charges inside a moving object. This separation is what creates an EMF, measurable as voltage across the car's sides.
  • Horizontal motion: direction the car moves within the field.
  • Key to understanding induced EMF.
Vertical Magnetic Field
A vertical magnetic field is one that has a direction either straight downward or upward with respect to the horizon. Earth's magnetic fields are not just horizontal, but include vertical components depending on location.
In this exercise, the vertical component is downward, with a strength of 4.8 x 10^{-5} T. This field interacts with the car as it moves horizontally. The interaction of the vertical field across the width of the car traveling forward leads to an induced voltage between the car鈥檚 sides.
Understanding vertical fields is essential as they're a determinant of the direction and magnitude of induced EMFs. They influence the way currents are generated in conductive materials and impact the induced voltage's behavior.
  • Induces effects when objects move perpendicularly.
  • Plays a major role in current direction and charge distribution.

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Most popular questions from this chapter

ssm A circular coil \((950 \text { turns, radius }=0.060 \mathrm{m})\) is rotating in a uniform magnetic field. At \(t=0\) s, the normal to the coil is perpendicular to the magnetic field. At \(t=0.010\) s, the normal makes an angle of \(\phi=45^{\circ}\) with the field because the coil has made one-eighth of a revolution. An average emf of magnitude 0.065 \(\mathrm{V}\) is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.

A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is \(45.0 \Omega .\) The area of each turn is \(4.70 \times 10^{-4} \mathrm{~m}^{2} .\) This coil is moved from a region where the magnetic field is zero into a region where it is nonzero, the normal to the coil being kept parallel to the magnetic field. The amount of charge that is induced to flow around the circuit is measured to be \(8.87 \times 10^{-3} \mathrm{C}\). Find the magnitude of the magnetic field.

A vacuum cleaner is plugged into a \(120.0-\mathrm{V}\) socket and uses 3.0 \(\mathrm{A}\) of current in normal operation when the back emf generated by the electric motor is 72.0 \(\mathrm{V}\) . Find the coil resistance of the motor.

In a television set the power needed to operate the picture tube comes from the secondary of a transformer. The primary of the trans- former is connected to a \(120-\mathrm{V}\) receptacle on a wall. The picture tube of the television set uses 91 \(\mathrm{W}\) , and there is 5.5 \(\mathrm{mA}\) of current in the secondary coil of the transformer to which the tube is connected. Find the turns ratio \(N_{s} / N_{\mathrm{p}}\) of the transformer.

In some places, insect "zappers," with their blue lights, are a familiar sight on a summer's night. These devices use a high voltage to electrocute insects. One such device uses an ac voltage of 4320 \(\mathrm{V}\) , which is obtained from a standard \(120.0-\mathrm{V}\) outlet by means of a transformer. If the primary coil has 21 turns, how many turns are in the secondary coil?
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