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Chapter 21: Problem 82

A particle has a charge of \(q=+5.60 \mu \mathrm{C}\) and is located at the coordinate origin. As the drawing shows, an electric field of \(E_{x}=+245 \mathrm{N} / \mathrm{C}\) exists along the \(+x\) axis, A magnetic field also exists, and its \(x\) and \(y\) components are \(B_{x}=+1.80 \mathrm{T}\) and \(B_{y}=+1.40 \mathrm{T}\) . Calculate the force (magnitude and direction) exerted on the particle by each of the three fields when it is \(\quad\) stationary, (b) moving along the \(+x\) axis at a speed of 375 \(\mathrm{m} / \mathrm{s}\) , and \(\quad\) (c) moving along the \(+z\) axis at a speed of 375 \(\mathrm{m} / \mathrm{s}\) .

Short Answer

Expert verified
(a) Electric force: 0.001372 N along +x; (b) Total force: 0.001372 N + 0.00294 N (in -z); (c) Total force: 0.00378 N (in -y).

Step by step solution

01

Identify the Forces When Stationary

When the particle is stationary, it only experiences the electric force, since magnetic forces act on moving charges. The electric force \( F_e \) is given by \( F_e = q \times E_x \). Plugging in the given values, we have \( F_e = (5.60 \times 10^{-6} \, \mathrm{C}) \times (245 \, \mathrm{N/C}) = 0.001372 \, \mathrm{N} \). The direction of the electric force is along the \(+x\) axis.
02

Calculate the Forces When Moving Along +x Axis

Since the particle is moving along the \(+x\) axis at \(375 \, \mathrm{m/s}\), it experiences both electric and magnetic forces. The electric force \( F_e \) remains the same as previously calculated. The magnetic force \( F_m \) is given by \( F_m = q \times v \times B_y \), where only the \( B_y \) component affects the force. \( F_m = (5.60 \times 10^{-6} \, \mathrm{C}) \times (375 \, \mathrm{m/s}) \times (1.40 \, \mathrm{T}) = 0.00294 \, \mathrm{N} \). By the right-hand rule, this force is directed in the negative \(z\) direction.
03

Calculate the Forces When Moving Along +z Axis

For motion along the \(+z\) axis, the electric force \( F_e \) remains unaffected by speed. The magnetic force \( F_m \) now depends on both \( B_x \) and the velocity \( v \). \( F_m = q \times v \times B_x = (5.60 \times 10^{-6} \, \mathrm{C}) \times (375 \, \mathrm{m/s}) \times (1.80 \, \mathrm{T}) = 0.00378 \, \mathrm{N} \). This force is directed along the negative \(y\) axis based on the right-hand rule.

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Key Concepts

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

Electric Fields
Electric fields are regions around charged particles or objects where other charges would experience a force. Imagine it as an invisible field surrounding a charged object. When a positive charge enters this field, it experiences a force in the direction of the field lines.
The strength of an electric field is expressed in newtons per coulomb (N/C).
It depends on the amount of charge that creates the field and the distance from the charge.
  • Electric fields exert forces on charges present within them.
  • The force felt by a charge in an electric field is given by: \( F_e = q \times E \), where \( q \) is the charge and \( E \) is the electric field strength.
This relationship helps explain how charges attract or repel each other depending on the field's direction.
Magnetic Fields
Magnetic fields, much like electric fields, are invisible areas around a magnet where magnetic forces can be felt. These fields are formed by the movement of electric charges, such as electrons. Magnetic field lines are closed loops that emerge from the north pole of a magnet and enter the south pole.

Magnetic fields have various characteristics:
  • The strength of a magnetic field is typically measured in teslas (T).
  • Unlike electric fields, magnetic fields do not exert forces on stationary charges.
  • However, once the charge starts moving, a magnetic force can act on it.
The direction of these forces is perpendicular to the direction of the magnetic field and the velocity of the charge. This interaction is the principle behind many electrical devices and scientific phenomena.
Lorentz Force
The Lorentz force is a fundamental concept in electromagnetism describing the force exerted on a charged particle moving through electric and magnetic fields. This force combines both the electric force and the magnetic force.

The Lorentz force equation is given by:
\[ F = q(E + v \times B) \]
where:
  • \( q \) is the charge of the particle,
  • \( E \) is the electric field strength,
  • \( v \) is the velocity of the particle, and
  • \( B \) is the magnetic field.
The formula above shows that the total force is the sum of the electric and magnetic forces.
Understanding this force is crucial for predicting how charged particles will move in varying electric and magnetic environments.
Right-Hand Rule
The right-hand rule is a handy mnemonic for determining the direction of the magnetic force exerted on a moving charge in a magnetic field. Although there are several variants of this rule, they all help visualize interactions between charge, velocity, and magnetic fields.

Here's how to use the right-hand rule:
  • Point your right thumb in the direction of the particle's velocity \( v \).
  • Extend your fingers in the direction of the magnetic field \( B \).
  • Your palm faces the direction of the force on a positive charge.
If the charge is negative, like an electron, the force direction would be opposite to your palm.
Remembering the right-hand rule allows visualization of the three-dimensional nature of magnetism and aids in problem-solving related to the Lorentz force.

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Most popular questions from this chapter

The \(1200-\) -turn coil in a dc motor has an area per turn of \(1.1 \times 10^{-2} \mathrm{m}^{2}\) The design for the motor specifies that the magnitude of the maximum torque is 5.8 \(\mathrm{N} \cdot \mathrm{m}\) when the coil is placed in a \(0.20-\mathrm{T}\) magnetic field What is the current in the coil?

mmh Two circular coils are concentric and lie in the same plane. The inner coil contains 140 turns of wire, has a radius of \(0.015 \mathrm{m},\) and carries a current of 7.2 \(\mathrm{A}\) . The outer coil contains 180 turns and has a radius of 0.023 \(\mathrm{m}\) . What must be the magnitude and direction (relative to the current in the inner coil) of the current in the outer coil, so that the net magnetic field at the common center of the two coils is zero?

You have a wire of length \(L=1.00 \mathrm{m}\) from which to make the square coil of a dc motor. The current in the coil is \(I=1.7 \mathrm{A},\) and the magnetic field of the motor has a magnitude of \(B=0.34 \mathrm{T}\) . Find the maximum torque exerted on the coil when the wire is used to make a single-turn square coil and a two-turn square coil.

ssm The magnetic field produced by the solenoid in a magnetic resonance imaging (MRI) system designed for measurements on whole human bodies has a field strength of \(7.0 \mathrm{T},\) and the current in the solenoid is 2.0 \(\times 10^{2}\) A. What is the number of turns per meter of length of the solenoid? Note that the solenoid used to produce the magnetic field in this type of system has a length that is not very long compared to its diameter. Because of this and other design considerations, your answer will be only an approximation.

The ion source in a mass spectrometer produces both singly and doubly ionized species, \(\mathrm{X}^{+}\) and \(\mathrm{X}^{2+}\) . The difference in mass between these species is too small to be detected. Both species are accelerated through the same electric potential difference, and both experience the same magnetic ficld, which causes them to move on circular paths. The radius of the path for the species \(X^{+}\) is \(r_{1},\) while the radius for species \(X^{2+}\) is \(r_{2}\) . Find the ratio \(r_{1} / r_{2}\) of the radii.
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