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Chapter 21: Problem 79

A charge is moving perpendicular to a magnetic field and experiences a force whose magnitude is \(2.7 \times 10^{-3} \mathrm{N}\) . If this same charge were to move at the same speed and the angle between its velocity and the same magnetic field were \(38^{\circ},\) what would be the magnitude of the magnetic force magnitude of the magnetic force that the charge would experience?

Short Answer

Expert verified
The new force is approximately \( 1.66 \times 10^{-3} \mathrm{N} \).

Step by step solution

01

Understanding the Magnetic Force Formula

The magnetic force experienced by a moving charge in a magnetic field is given by the equation \( F = qvB \sin(\theta) \) where \( F \) is the force, \( q \) is the charge, \( v \) is the velocity of the charge, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the velocity and the magnetic field. For a perpendicular motion, \( \theta = 90^{\circ} \), so \( \sin(90^{\circ}) = 1 \).
02

Calculate the Force when Perpendicular

When the charge moves perpendicular to the magnetic field (\( \theta = 90^{\circ} \)), the force is \( F = qvB \). The problem states that the magnitude of this force is \( 2.7 \times 10^{-3} \mathrm{N} \). This is our reference force when \( \sin(90^{\circ}) = 1 \).
03

Calculate the Force with Angle 38掳

Now the charge moves at an angle of \( 38^{\circ} \) to the magnetic field. Thus the force formula becomes \( F' = qvB \sin(38^{\circ}) \). To find this force \( F' \), we use the ratio \( \frac{F'}{F} = \frac{\sin(38^{\circ})}{\sin(90^{\circ})} \). Since \( \sin(90^{\circ}) = 1 \), we have \( F' = F \cdot \sin(38^{\circ}) \).
04

Solve for the New Force

Substitute \( F = 2.7 \times 10^{-3} \mathrm{N} \) and calculate \( F' \). Using \( \sin(38^{\circ}) \approx 0.6157 \), calculate the new force: \[ F' = 2.7 \times 10^{-3} \times 0.6157 = 1.66239 \times 10^{-3} \mathrm{N} \]. Thus, the new force experienced by the charge is approximately \( 1.66 \times 10^{-3} \mathrm{N} \).

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Key Concepts

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

Moving Charge
When a charge moves through space, it creates a magnetic force if there's a magnetic field present. Think of a moving charge like a small current, and currents naturally interact with magnetic fields. The charge's movement, or velocity, plays a key role here. If you increase the speed of the charge, the magnetic interaction becomes stronger.

Here's why this is crucial:
  • Moving charges are the sole reason magnetic forces exist in this context. Without movement, there's no magnetic force from the charge itself.
  • The velocity's direction gives the whole scenario a directional dependency, linking closely to how the magnetic field affects the charge.
Understanding how a charge can create and interact with magnetic forces provides insight into various technologies, like MRI machines and electric motors.
Magnetic Field
A magnetic field is an invisible area of influence exerted by magnetic materials or moving charges. Think of it as a map that tells charged particles how to move in space. When a moving charge enters a magnetic field, it experiences a force due to this invisible influence. The stronger the magnetic field (represented by the variable \( B \)), the greater the force on the charge.

Important aspects of magnetic fields:
  • Magnetic field lines illustrate the strength and direction of the field. Close lines mean a stronger field.
  • The magnetic field is measured in Tesla (T), and typical values depend on the application or material generating it.
Magnetic fields influence many everyday phenomena, from how compasses work to the storage of data in computers.
Angle of Motion
The angle of motion between a charge's velocity and a magnetic field determines the size of the force experienced by the charge. In the equation \( F = qvB \sin(\theta) \), \( \theta \) is the angle we're interested in. When \( \theta = 90^{\circ} \), the charge moves perpendicular to the magnetic field, creating the maximum force.

Here's a breakdown:
  • A 0-degree angle means the charge moves in the same direction as the magnetic field, resulting in no force.
  • At 90 degrees, the movement is completely opposed to the field lines, causing the greatest interaction.
  • Any angle in between reduces the force by a scale related to \( \sin(\theta) \).
Understanding this angle helps us work with devices ranging from power tools to sophisticated scientific equipment.
Sine Function
The sine function is a crucial mathematical tool when dealing with angles, especially in magnetic force calculations. In the formula \( F = qvB \sin(\theta) \), \( \sin(\theta) \) helps adjust the force based on the angle between the charge's motion and the magnetic field.

Why does \( \sin(\theta) \) matter?
  • It acts as a "weighting factor" that scales the force depending on how aligned the charge's velocity and the magnetic field are.
  • \( \sin(0^{\circ}) = 0 \) means no force; \( \sin(90^{\circ}) = 1 \) gives full force.
  • The sine function itself varies smoothly between 0 and 1 as the angle changes from 0 to 90 degrees, allowing a continuous range of force magnitudes.
Sine functions are essential in physics and engineering, providing an intuitive way to model waveforms, oscillations, and rotations.

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Most popular questions from this chapter

A horizontal wire is hung from the ceiling of a room by two massless strings. The wire has a length of 0.20 \(\mathrm{m}\) and a mass of 0.080 kg. A uniform magnetic field of magnitude 0.070 \(\mathrm{T}\) is directed from the ceiling to the floor.When a current of \(I=42 \mathrm{A}\) exists in the wire, the wire swings upward and, at equilibrium, makes an angle \(\phi\) with respect to the vertical, as the drawing shows. Find (a) the angle \(\phi\) and \((b)\) the tension in each of the two strings.

mmh A very long, hollow cylinder is formed by rolling up a thin sheet of copper. Electric charges flow along the copper sheet parallel to the axis of the cylinder. The arrangement is, in effect, a hollow tube of current I. Use Ampere's law to show that the magnetic field (a) is \(\mu_{0} I /(2 \pi r)\) outside the cylinder at a distance \(r\) from the axis and (b) is zero at any point within the hollow interior of the cylinder. (Hint: For closed paths, use circles perpendicular to and centered on the axis of the cylinder.

ssm In the operating room, anesthesiologists use mass spectrometers to monitor the respiratory gases of patients undergoing surgery. One gas that is often monitored is the anesthetic isoflurane (molecular mass \(=3.06 \times 10^{-25} \mathrm{kg} ) .\) In a spectrometer, a singly ionized molecule of isoflurane (charge \(=+e )\) moves at a speed of \(7.2 \times 10^{3} \mathrm{m} / \mathrm{s}\) /s on a circular path that hat has a radius of 0.10 \(\mathrm{m}\) . What is the magnitude of the magnetic field that the spectrometer uses?

In a certain region, the carth's magnetic ficld has a magnitude of \(5.4 \times 10^{-5} \mathrm{T}\) and is directed north at an angle of \(58^{\circ}\) below the horizontal. An electrically charged bullet is fired north and \(11^{\circ}\) above the horizontal, with a speed of 670 \(\mathrm{m} / \mathrm{s}\) . The magnetic force on the bullet is \(2.8 \times 10^{-10} \mathrm{N}\) directed due east. Determine the bullet's electric charge, including its algebraic sign \((+\text { or }-) .\)

A charged particle with a charge-to-mass ratio of \(|q| / m=5.7 \times 10^{8} \mathrm{C} / \mathrm{kg}\) travels on a circular path that is perpendicular to a magnetic field whose magnitude is 0.72 \(\mathrm{T}\) . How much time does it take for the particle to complete one revolution?
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