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Ch. 28 - Sources of Magnetic Field
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 28 - Sources of Magnetic FieldProblem 19
Chapter 27, Problem 19

(II) Let two long parallel wires, a distance d apart, carry equal dc currents I in the same direction. One wire is at 𝓍 = 0, the other at 𝓍 = d, Fig. 28–41. Determine B→\(\overrightarrow{B}\) along the 𝓍 axis between the wires as a function of 𝓍.

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Step 1: Understand the problem. Two long parallel wires separated by a distance d carry equal currents I in the same direction. We are tasked with finding the magnetic field B⃗ along the x-axis between the wires as a function of x, where x is the position along the axis between the wires.
Step 2: Recall the formula for the magnetic field produced by a long straight current-carrying wire at a distance r from the wire. The magnetic field is given by: B = μ₀I2πr, where μ₀ is the permeability of free space, I is the current, and r is the distance from the wire.
Step 3: Consider the contributions to the magnetic field at a point x along the x-axis from both wires. The wire at x = 0 produces a magnetic field at x due to the current I, and the wire at x = d also produces a magnetic field at x. Use the right-hand rule to determine the direction of the magnetic fields from each wire.
Step 4: Write the expressions for the magnetic fields from each wire. For the wire at x = 0, the distance to the point x is |x|, so the magnetic field is: B1 = μ₀I2πx. For the wire at x = d, the distance to the point x is |d - x|, so the magnetic field is: B2 = μ₀I2π|d-x|.
Step 5: Combine the magnetic field contributions. Since the currents are in the same direction, the magnetic fields from the two wires will oppose each other along the x-axis. The net magnetic field at a point x is the difference between the magnitudes of the two fields: B = |B1 - B2|. Substitute the expressions for B₁ and B₂ to find the net magnetic field as a function of x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Field Due to a Current-Carrying Wire

A long straight wire carrying a current generates a magnetic field around it, described by Ampère's Law. The magnetic field strength (B) at a distance (r) from the wire is given by the formula B = (μ₀I)/(2πr), where μ₀ is the permeability of free space and I is the current. The direction of the magnetic field can be determined using the right-hand rule, which states that if you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field.
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Superposition of Magnetic Fields

When multiple sources generate magnetic fields, the total magnetic field at a point is the vector sum of the individual fields. In this scenario, the magnetic fields produced by each wire will interact, and their contributions must be added together to find the resultant magnetic field at any point along the x-axis. This principle is crucial for determining the net magnetic field between the two wires, as the fields will have both magnitude and direction.
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Distance and Magnetic Field Variation

The magnetic field strength varies with distance from the current-carrying wires. As you move along the x-axis between the two wires, the distance from each wire changes, affecting the magnetic field contributions from each. Specifically, the magnetic field will be stronger closer to the wire and weaker as you move away, which must be accounted for when calculating the total magnetic field at any given point between the wires.
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Related Practice
Textbook Question

(II) Two long parallel wires 8.20 cm apart carry 19.5-A dc currents in the same direction. Determine the magnetic field vector at a point P, 12.0 cm from one wire and 13.0 cm from the other. See Fig. 28–43. [Hint: Use the law of cosines. See Appendix A or inside rear cover.]

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Textbook Question

(II) An electron enters a uniform magnetic field B = 0.28 T at a 45° angle to B→\(\overrightarrow{B}\). Determine the radius r and pitch p (distance between loops) of the electron’s helical path assuming its speed is 2.2 x 106 m/s. See Fig. 27–48.


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Textbook Question

"(II) A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 28–40. There is a dc current of 3.5 A in both wires. Determine the magnitude and direction of the net force on the loop.


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Textbook Question

(II) Two long straight wires each carry a dc current I out of the page toward the viewer, Fig. 28–38. Indicate, with appropriate arrows, the direction of B→\(\overrightarrow{B}\) at each of the points 1 to 6 in the plane of the page. State if the field is zero at any of the points.

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Textbook Question

(II) Two long thin parallel wires 13.0 cm apart carry 25-A currents in the same direction. Determine the magnetic field vector at a point 10.0 cm from one wire and 6.0 cm from the other (Fig. 28–37). [Hint: You could try using the law of cosines, Appendix A.]

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Textbook Question

(II) Two long wires are oriented so that they are perpendicular to each other. At their closest, they are 20.0 cm apart (Fig. 28–42). What is the magnitude of the magnetic field at a point midway between them if the top one carries a current of 18.0 A and the bottom one carries 12.0 A?

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