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Chapter 21: Problem 31

A wire carries a current of 0.66 A. This wire makes an angle of \(58^{\circ}\) with respect to a magnetic field of magnitude \(4.7 \times 10^{-5} \mathrm{~T}\). The wire experiences a magnetic force of magnitude \(7.1 \times 10^{-5} \mathrm{~N}\). What is the length of the wire?

Short Answer

Expert verified
The length of the wire is approximately 2.399 meters.

Step by step solution

01

Identify the Formula

To find the length of the wire, we use the formula for the magnetic force on a current-carrying wire: \[ F = I \cdot B \cdot L \cdot \sin(\theta) \]where:- \( F \) is the magnetic force, - \( I \) is the current, - \( B \) is the magnetic field,- \( L \) is the length of the wire,- \( \theta \) is the angle between the wire and the magnetic field.
02

Rearrange the Formula for Length

To solve for \( L \), rearrange the formula:\[ L = \frac{F}{I \cdot B \cdot \sin(\theta)} \]
03

Insert Known Values

Plug the given numerical values into the formula:- \( F = 7.1 \times 10^{-5} \; \mathrm{N} \)- \( I = 0.66 \; \mathrm{A} \)- \( B = 4.7 \times 10^{-5} \; \mathrm{T} \)- \( \theta = 58^{\circ} \)Thus, we have:\[ L = \frac{7.1 \times 10^{-5}}{0.66 \times 4.7 \times 10^{-5} \times \sin(58^{\circ})} \]
04

Calculate \( \sin(58^{\circ}) \)

Calculate the sine of the angle:\[ \sin(58^{\circ}) \approx 0.848 \]
05

Compute the Length of the Wire

Substitute \( \sin(58^{\circ}) \approx 0.848 \) into the equation:\[ L = \frac{7.1 \times 10^{-5}}{0.66 \times 4.7 \times 10^{-5} \times 0.848} \]Calculate the numerical value to find \( L \).

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Key Concepts

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

Current-Carrying Wire
A current-carrying wire is essential in creating a magnetic field around it when an electric current flows through. This arises from the fundamental relationship between electricity and magnetism, known as electromagnetism. When current, measured in amperes (A), passes through a wire, it generates a magnetic field surrounding the wire.

In practical scenarios, this principle is utilized in various applications, including electromagnets, transformers, and electric motors. Understanding how a wire carrying current behaves in a magnetic field is vital, especially when calculating forces acting on the wire.
  • The current intensity dictates the strength of the magnetic field generated.
  • The direction of the field lines is perpendicular to the current.
The interplay between current and magnetic forces can significantly affect the wire's behavior, influencing its movement and interactions with nearby magnetic materials.
Magnetic Field
A magnetic field refers to the space around a magnet or current-carrying conductor where magnetic forces are observed. Its strength and direction can affect materials or charges moving through it, especially wires with electric currents. The intensity of a magnetic field is measured in teslas (T).

Magnetic fields are essential in physics as they interact with charged particles, including electron streams in wires, leading to forces that change velocities or directions. In the context of a current-carrying wire, understanding this interaction helps determine the resulting forces.
  • Magnetic forces on a wire depend on the wire's length and current, as well as the magnetic field strength.
  • A stronger magnetic field results in a more significant force, affecting the wire.
This principle is crucial in engineering systems relying on magnetic force applications.
Sine Function
The sine function is a fundamental trigonometric function describing the ratio of the length of the side opposite to an angle to the hypotenuse in a right triangle. It is especially relevant in calculating components perpendicular to one another, such as forces.

In this context, the sine function determines how much of the force acts in a specific direction when a wire makes an angle within a magnetic field. Accurate calculations rely on understanding how sine values change with different angles, expressed in degrees or radians.
  • The sine value ranges from -1 to 1, representing force directions optimally.
  • A sine of 0 signifies parallel orientation, while 1 or -1 indicates perpendicular orientation.
Mastering the sine function is crucial in physics for predicting and calculating vector quantities like force in mixed dimensional scenarios.
Angle of Inclination
The angle of inclination denotes the angle between a wire and lines of a magnetic field. Inclination angles are measured in degrees or radians, influencing how forces are computed in association with the magnetic field.

When a wire is inclined relative to the magnetic field, the angle affects the resultant force experienced by the wire. This force depends on both the magnitude of the magnetic force and the angle's sine value.
  • An angle of 0 indicates that the forces align parallel, minimizing interaction.
  • A substantial angle, close to 90 degrees, maximizes the interaction due to the perpendicular force component.
Thus, recognizing the impact of inclination angles is fundamental in solving complex physical problems involving magnetic fields and forces.

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Most popular questions from this chapter

A wire has a length of \(7.00 \times 10^{-2} \mathrm{~m}\) and is used to make a circular coil of one turn. There is a current of \(4.30 \mathrm{~A}\) in the wire. In the presence of a 2.50-T magnetic field, what is the maximum torque that this coil can experience?

Two charged particles move in the same direction with respect to the same magnetic field. Particle 1 travels three times faster than particle 2 . However, each particle experiences a magnetic force of the same magnitude. Find the ratio \(\left|q_{1}\right| /\left|q_{2}\right|\) of the magnitudes of the charges.

The maximum torque experienced by a coil in a 0.75 -T magnetic field is \(8.4 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m}\). The coil is circular and consists of only one turn. The current in the coil is \(3.7 \mathrm{~A}\). What is the length of the wire from which the coil is made?

A positively charged particle of mass \(7.2 \times 10^{-8} \mathrm{~kg}\) is traveling due east with a speed of \(85 \mathrm{~m} / \mathrm{s}\) and enters a 0.31 -T uniform magnetic field. The particle moves through onequarter of a circle in a time of \(2.2 \times 10^{-3} \mathrm{~s}\), at which time it leaves the field heading due south. All during the motion the particle moves perpendicular to the magnetic field. (a) What is the magnitude of the magnetic force acting on the particle? (b) Determine the magnitude of its charge.

Two circular loops of wire, each containing a single turn, have the same radius of \(4.0 \mathrm{~cm}\) and a common center. The planes of the loops are perpendicular. Each carries a current of 1.7 A. What is the magnitude of the net magnetic field at the common center?
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