Ch. 27 - Magnetism

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 27 - Magnetism

Problem 7

Chapter 26, Problem 7

The magnetic force per meter on a wire is measured to be only 55% of its maximum possible value. What is the angle between the wire and the magnetic field?

Verified step by step guidance

The magnetic force on a current-carrying wire in a magnetic field is given by the formula:

F = I L B sin θ

, where

F

is the magnetic force,

I

is the current,

L

is the length of the wire,

B

is the magnetic field strength, and

θ

is the angle between the wire and the magnetic field.

The maximum possible magnetic force occurs when

sin θ = 1

, which corresponds to

θ = 90 °

. In this case, the force is

F = I L B

In the problem, the magnetic force is only 55% of its maximum value. This means

sin θ = 0.55

To find the angle

θ

, take the inverse sine (arcsin) of 0.55:

θ = arcsin (0.55)

Use a calculator or mathematical tool to compute the value of

θ

in degrees or radians, depending on the context of the problem.

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Key Concepts

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

Magnetic Force on a Current-Carrying Wire

The magnetic force experienced by a wire carrying an electric current in a magnetic field is given by the formula F = I * L * B * sin(θ), where F is the force, I is the current, L is the length of the wire, B is the magnetic field strength, and θ is the angle between the wire and the magnetic field. This relationship shows that the force is maximized when the wire is perpendicular to the magnetic field (θ = 90°) and is zero when parallel (θ = 0°).

Maximum Magnetic Force

The maximum magnetic force occurs when the angle θ is 90 degrees, resulting in sin(θ) being equal to 1. In this scenario, the force is at its peak value, which can be calculated using the formula F_max = I * L * B. Understanding this concept is crucial for determining the actual angle when the force is less than the maximum possible value.

Trigonometric Relationships

Trigonometric functions, particularly the sine function, play a vital role in relating angles to ratios in right triangles. In the context of magnetic forces, the sine of the angle θ determines the proportion of the maximum force that is realized. If the magnetic force is 55% of its maximum, this can be expressed mathematically as sin(θ) = 0.55, allowing for the calculation of the angle using the inverse sine function.

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Related Practice

Textbook Question

(II) A current-carrying circular loop of wire (radius r, current I) is partially immersed in a magnetic field of constant magnitude B₀ directed out of the page as shown in Fig. 27–43. Determine the net force on the loop due to the field in terms of θ₀. (Note that θ₀ points to the dashed line, above which B = 0.)

Textbook Question

A stiff wire 50.0 cm long is bent at a right angle in the middle. One section lies along the z axis and the other is along the line y = 2x in the xy plane. A current of 20.0 A flows in the wire—down the z axis and out the wire in the xy plane. The wire passes through a uniform magnetic field given by = (0.285î ) T. Determine the magnitude and direction of the total force on the wire.

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Textbook Question

(III) A curved wire, connecting two points a and b, lies in a plane perpendicular to a uniform magnetic field $\vec{B}$ and carries a current I. Show that the resultant magnetic force on the wire, no matter what its shape, is the same as that on a straight wire connecting the two points carrying the same current I. See Fig. 27–44.

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Textbook Question

(a) What is the force per meter of length on a straight wire carrying a 7.40-A current when perpendicular to a 0.90-T uniform magnetic field?

(b) What if the angle between the wire and field is 35.0°?

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