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Chapter 20: Problem 42

\(\bullet\) (a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 \(\mathrm{cm}\) from the wire is equal to 1.00 \(\mathrm{G}\) (comparable to the earth's northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth's magnetic field? (c) Repeat part (b) except with the wire vertical and the current going upward.

Short Answer

Expert verified
(a) 10 A; (b) South of the wire; (c) East of the wire.

Step by step solution

01

Understand the Formula

The magnetic field around a long straight wire is given by Amp猫re's law: \( B = \frac{\mu_0 I}{2\pi r} \), where \( B \) is the magnetic field, \( \mu_0 = 4\pi \times 10^{-7} \, T\cdot m/A \) is the permeability of free space, \( I \) is the current, and \( r \) is the distance from the wire.
02

Solve for Current

Given \( B = 1.00 \, G = 1.00 \times 10^{-4} \, T \) and \( r = 2.00 \, cm = 0.02 \, m \), substitute these into the formula: \( 1.00 \times 10^{-4} = \frac{4\pi \times 10^{-7} I}{2\pi \times 0.02} \). Simplify to solve for \( I \): \( I = \frac{1.00 \times 10^{-4} \times 2\pi \times 0.02}{4\pi \times 10^{-7}} = 10 \, A \).
03

Analyze Horizontal Wire

For a horizontal wire carrying current from east to west, use the right-hand rule to determine the direction of the magnetic field: with the thumb pointing in the direction of the current (east-west), the magnetic field circles the wire. To match the earth's northward field, positions directly south of the wire will have a northward field component.
04

Analyze Vertical Wire

For a vertical wire, current going upwards, apply the right-hand rule again. If you wrap your right hand around the wire with your thumb pointing upwards, the horizontal field will point clockwise around the wire. The northward component would appear on the eastern side of the wire, corresponding with the earth's northward field.

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Key Concepts

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

Amp猫re's Law
Amp猫re's Law is a cornerstone in the study of magnetism, particularly for understanding the magnetic field around current-carrying wires. It states that the magnetic field (B ) produced by an electric current in a conductor is proportional to the current (I ) and inversely proportional to the distance (r) from the wire. Amp猫re's Law can be expressed with the formula: \( B = \frac{\mu_0 I}{2\pi r} \), where:
  • \( \mu_0 \) is the permeability of free space, valued at \( 4\pi \times 10^{-7} \, T\cdot m/A \).
  • \( I \) is the current in amperes.
  • \( r \) is the distance from the wire in meters.
This formula reveals how the magnetic field strength decreases with increased distance from the wire. By manipulating the equation, you can calculate the necessary current to produce a desired magnetic field strength at a given distance. Understanding this law is essential for solving problems related to magnetic fields in circuits and various electrical applications.
Right-Hand Rule
The Right-Hand Rule is a mnemonic used to determine the direction of a magnetic field around a current-carrying conductor. It's a simple yet essential technique that enhances spatial reasoning when dealing with electromagnetic fields. To apply the Right-Hand Rule, follow these steps:
  • Position your right hand such that your thumb points in the direction of the current flow.
  • Your other fingers will naturally curl, indicating the direction of the magnetic field lines encircling the wire.
When you encounter scenarios like a horizontal wire with current traveling from east to west, using your right hand ensures you accurately predict the magnetic field's orientation. This tool is especially useful when aligning with or against other magnetic fields, such as the earth's magnetic field, allowing for a clear and intuitive understanding of magnetic interactions.
Current and Magnetic Fields
The relationship between electric current and magnetic fields is fundamental to electromagnetism. When electric current passes through a conductor, it generates a magnetic field perpendicular to the flow of the current. In practice:
  • A straight wire carrying a direct current produces circular magnetic field lines around it, as predicted by Amp猫re's Law.
  • The Right-Hand Rule helps ascertain the direction these field lines take relative to the current direction.
For instance, with a horizontal wire carrying current east to west, the field lines create circles around the wire. The location of significant horizontal magnetic components can be predicted by examining the position relative to the earth's magnetic north. Similarly, a vertical wire with upward current creates a magnetic field that loops around clockwise when viewed from above. Understanding current and magnetic fields extends beyond isolated experiments. It forms the basis for various technologies and engineering solutions, such as electric motors, transformers, and magnetic storage devices.

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Most popular questions from this chapter

At a given instant, a particle with a mass of \(5.00 \times 10^{-3} \mathrm{kg}\) and a charge of \(3.50 \times 10^{-8} \mathrm{C}\) has a velocity with a magnitude of \(2.00 \times 10^{5} \mathrm{m} / \mathrm{s}\) in the \(+y\) direction. It is moving in a uniform magnetic field that has magnitude 0.8 \(\mathrm{T}\) and is in the \(-x\) direction. What are (a) the magnitude and direction of the magnetic force on the particle and (b) its resulting acceleration?

A solenoid that is 35 \(\mathrm{cm}\) long and contains 450 circular coils 2.0 \(\mathrm{cm}\) in diameter carries a 1.75 A current. (a) What is the magnetic field at the center of the solenoid, 1.0 \(\mathrm{cm}\) from the coils? (b) Suppose we now stretch out the coils to make a very long wire carrying the same current as before. What is the magnetic field 1.0 \(\mathrm{cm}\) from the wire's center? Is it the same as you found in part (a)? Why or why not?

\bullet Bubble chamber, I. Certain types of bubble chambers are filled with liquid hydrogen. When a particle (such as an electron or a proton) passes through the liquid, it leaves a track of bubbles, which can be photographed to show the path of the particle. The apparatus is immersed in a known magnetic field, which causes the particle to curve. Figure 20.77 is a trace of a bubble chamber image showing the path of an electron. (a) How could you determine the sign of the charge of a particle from a photograph of its path? (b) How can physicists determine the momentum and the speed of this electron by using measurements made on the photograph, given that the magnetic field is known and is perpendicular to the plane of the figure? (c) The electron is obviously spiraling into smaller and smaller circles. What properties of the electron must be changing to cause this behavior? Why does this happen? (d) What would be the path of a neutron in a bubble chamber? Why?

An ion having charge \(+6 e\) is traveling horizontally to the left at 8.50 \(\mathrm{km} / \mathrm{s}\) when it enters a magnetic field that is perpendicular to its velocity and deflects it downward with an initial magnetic force of \(6.94 \times 10^{-15} \mathrm{N} .\) What are the direction and magnitude of this field? Illustrate your method of solving this problem with a diagram.

\(\bullet\) In a 1.25 T magnetic field directed vertically upward, a particle having a charge of magnitude 8.50\(\mu \mathrm{C}\) and initially moving northward at 4.75 \(\mathrm{km} / \mathrm{s}\) is deflected toward the east. (a) What is the sign of the charge of this particle? Make a sketch to illustrate how you found your answer. (b) Find the magnetic force on the particle.
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