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Chapter 20: Problem 26

A straight vertical wire carries a current of 1.20 \(\mathrm{A}\) down- ward in a region between the poles of a large electromagnet where the field strength is 0.588 \(\mathrm{T}\) and is horizontal. What are the magnitude and direction of the magnetic force on a 1.00 \(\mathrm{cm}\) section of this wire if the magnetic-field direction is (a) toward the east, (b) toward the south, (c) \(30.0^{\circ}\) south of west?

Short Answer

Expert verified
(a) 0.007056 N into the page; (b) 0.007056 N west; (c) 0.006102 N into the page, slightly northwest.

Step by step solution

01

Understand the Formula

The magnetic force on a current-carrying wire in a magnetic field can be calculated using the formula \( F = I \cdot L \cdot B \cdot \sin(\theta) \), where \( F \) is the force, \( I \) is the current, \( L \) is the length of the wire, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the direction of the current and the direction of the magnetic field.
02

Convert Length to Meters

The problem gives the length of the wire as 1.00 cm. Convert this to meters: \( L = 1.00 \ cm = 0.010 \ m \).
03

Calculate Force for Magnetic Field Toward East

For part (a), with the magnetic field toward the east, the current is downward, making a \( 90^{\circ} \) angle with the field (as east and downward are perpendicular). Hence \( \sin(\theta) = \sin(90^{\circ}) = 1 \). Substitute values into the formula: \( F = 1.20 \ A \times 0.010 \ m \times 0.588 \ T \times 1 = 0.007056 \ N \). Since the right-hand rule gives a force direction into the page, the force's magnitude is 0.007056 N directed into the page.
04

Calculate Force for Magnetic Field Toward South

For part (b), the magnetic field is towards the south, and the current is downward, also making a \( 90^{\circ} \) angle. The force is calculated similarly: \( F = 1.20 \ A \times 0.010 \ m \times 0.588 \ T \times 1 = 0.007056 \ N \). Using the right-hand rule, the force is directed west.
05

Calculate Force for Magnetic Field at 30 Degrees South of West

For part (c), the angle between downward current and the magnetic field, which is 30 degrees south of west, is \( 90^{\circ} + 30^{\circ} = 120^{\circ} \). Use the sine component: \( \sin(120^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \). The force is: \( F = 1.20 \ A \times 0.010 \ m \times 0.588 \ T \times \frac{\sqrt{3}}{2} = 0.006102 \ N \). Using the right-hand rule, the force direction is into the page and slightly northwest.

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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 an essential concept in electromagnetism. When an electric current flows through a wire, electric charges move through the conductor. These moving charges create a magnetic field around the wire.
The direction of the current is conventionally taken from positive to negative, which helps in determining the direction of the magnetic field created by the wire. The interaction of this magnetic field with external magnetic fields can result in a force on the wire.
  • Current (I): Measured in amperes, represents the flow of electric charge.
  • Length of wire (L): A crucial factor in force calculations, always measure in meters.
  • Direction: Changing the direction of current flow can alter the magnetic effects around the wire.
Understanding these principles is foundational for solving problems involving magnetic forces on wires.
Magnetic Field Strength
Magnetic field strength, often denoted by the letter \( B \), is a measure of the magnetic field's strength in a given area. It is measured in teslas (T). The stronger the magnetic field, the greater the force it can exert on a current-carrying wire.
An external magnetic field can be generated by magnets or electromagnets and can interact with the magnetic field produced by an electric current.
When calculating magnetic forces, the strength of the magnetic field is directly proportional to the force experienced by the wire:
  • Interaction: The magnetic field interacts with any nearby magnetic fields, leading to a measurable force.
  • Impact on Force: Greater field strength results in a larger force on the wire.
Understanding magnetic field strength is crucial when working with electromagnetic applications, such as motors and generators.
Right-Hand Rule
The right-hand rule is a simple method used to determine the direction of the magnetic force acting on a current-carrying wire. This rule helps visualize the direction in which the force will act.
This can be understood as follows:
  • Position your right hand such that your thumb points along the direction of the current.
  • Your fingers should point in the direction of the magnetic field.
  • Your palm then faces the direction of the magnetic force.
This rule provides an easy way to predict how wires will behave in magnetic fields without needing to perform complex calculations or simulations.
The right-hand rule is valuable in practical applications such as determining the motion of charged particles in magnetic fields.
Trigonometric Functions in Physics
Trigonometric functions, like sine and cosine, are instrumental in physics for calculating forces in systems where directions are not aligned. In problems involving angles, trigonometric functions can help determine the component of a force acting in a specific direction.
Sine (\( \sin \)) is particularly important in problems involving magnetic forces, especially when the angle \( \theta \) between vectors is involved.
  • The sine of 90 degrees is 1, resulting in maximum force.
  • At 0 degrees, sine is 0, leading to no force.
  • For other angles, \( \sin(\theta) \) modifies the effective force calculation depending on the angle of interaction.
Utilizing trigonometric functions properly allows for accurate predictions and assessments of forces in physics problems.
Electromagnetism Formula
The electromagnetism formula used to calculate the magnetic force on a current-carrying wire is pivotal: \[ F = I \cdot L \cdot B \cdot \sin(\theta) \].
This formula quantifies how current (\( I \)), wire length (\( L \)), magnetic field strength (\( B \)), and the angle \( \theta \) between the current and magnetic field interact to determine force magnitude.
  • Variables: Each component of the formula plays a specific role in determining the overall force.
  • Applications: This formula forms the basis for calculating forces in electromagnets, motors, and other devices.
  • Angle Relation: The angle allows for adjustment in force based on the directional relationship between current and field.
Mastering this formula is crucial for anyone studying electromagnetism as it is the foundation for analyzing and understanding the behavior of wires in magnetic environments.

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Most popular questions from this chapter

\(\bullet\) 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.

\(\bullet\) Between the poles of a powerful magnet is a cylindrical uniform magnetic field with a diameter of 3.50 \(\mathrm{cm}\) and a strength of 1.40 \(\mathrm{T}\) . A wire carries a current through the center of the field at an angle of \(65.0^{\circ}\) to the magnetic field lines. If the wire experiences a magnetic force of \(0.0514 \mathrm{N},\) what is the current flowing in it?

\(\cdot\) A solenoid contains 750 coils of very thin wire evenly wrapped over a length of 15.0 \(\mathrm{cm} .\) Each coil is 0.800 \(\mathrm{cm}\) in diameter. If this solenoid carries a current of \(7.00 \mathrm{A},\) what is the magnetic field at its center?

The effect of transmission lines. Two hikers are reading a compass under an overhead transmission line that is 5.50 \(\mathrm{m}\) above the ground and carries a current of 0.800 \(\mathrm{kA}\) in a horizontal direction from north to south. (a) Find the magnitude and direction of the magnetic field at a point on the ground directly under the transmission line. (b) One hiker suggests that they walk 50 \(\mathrm{m}\) away from the lines to avoid inaccurate compass readings due to the current. Considering that the earth's magnetic field is on the order of \(0.5 \times 10^{-4} \mathrm{T},\) is the current really a problem?

A rectangular 10.0 \(\mathrm{cm}\) by 20.0 \(\mathrm{cm}\) circuit carrying an 8.00 \(\mathrm{A}\) current is oriented with its plane parallel to a uniform 0.750 T magnetic field (Figure 20.62\()\) . (a) Find the magnitude and direction of the magnetic force on each segment \((a b, b c,\) etc. \()\) of this circuit. Illustrate your answers with clear diagrams. (b) Find the magnitude of the net force on the entire circuit.
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