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Chapter 27: Problem 14

An electron is projected vertically upward with a speed of \(1.70 \times 10^{6} \mathrm{~m} / \mathrm{s}\) into a uniform magnetic field of \(0.480 \mathrm{~T}\) that is directed horizontally away from the observer. Describe the electron's path in this field.

Short Answer

Expert verified
The electron follows a clockwise circular path perpendicular to the magnetic field and velocity direction.

Step by step solution

01

Understanding the Force on the Electron

The magnetic force acting on a moving charge in a magnetic field can be given by the formula: \( \vec{F} = q ( \vec{v} \times \vec{B} ) \), where \( \vec{v} \) is the velocity vector, \( \vec{B} \) is the magnetic field vector, and \( q \) is the charge of the particle. For an electron, \( q = -1.60 \times 10^{-19} \) C. The cross product indicates that the force will be perpendicular to both the velocity and the magnetic field.
02

Determining the Direction of Forces Using Right-Hand Rule

To determine the direction of the force, we use the right-hand rule: point your fingers in the direction of the electron's velocity (upward) and curl them towards the direction of the magnetic field (horizontally away). The thumb points towards the direction of the cross product. Since the electron has a negative charge, the force direction is opposite, i.e., toward the observer and to the left.
03

Analyzing the Nature of the Electron's Path

The magnetic force acts perpendicular to the electron's velocity, which implies it does not do work on the electron (changing its speed), rather, it alters only the direction of its velocity. This results in a circular motion. Given the force direction found in Step 2, the electron will move in a clockwise circular path as seen from above.
04

Describe Electron's Path

In a uniform magnetic field, the electron will move in a circular motion perpendicular to both the field lines and the initial velocity. The electron's path is a circular orbit in a plane perpendicular to both the upward velocity and the horizontal magnetic field line.

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Key Concepts

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

Electron Motion
When an electron moves through a magnetic field, its path is influenced by the Lorentz force. This force acts on the electron due to its charge and motion within the magnetic field. The electron's initial velocity is crucial in determining how it interacts with the field. In our example, the electron is moving vertically upward with a speed of \(1.70 \times 10^{6} \mathrm{~m/s}\). This velocity vector guides how the magnetic field affects the path of the electron. The motion of the electron is not straight, but involves a force acting perpendicular to its velocity. This force leads to continuous change in direction, while the electron maintains its speed.
Right-Hand Rule
The right-hand rule is a simple tool used to find the direction of the magnetic force on a charged particle, like an electron, in a magnetic field. Here鈥檚 how it works:
  • Point your fingers in the direction of the electron鈥檚 velocity (upward in this case).
  • Curl your fingers in the direction of the magnetic field (horizontally away from you).
  • Your thumb points in the direction of the magnetic force.
However, because electrons are negatively charged, the actual force on the electron is in the opposite direction to your thumb. In our specific situation, this means the force on the electron is directed toward the observer and slightly to the left. This reversal is an essential part of using the right-hand rule for negative charges.
Circular Motion
In the presence of a magnetic field, the motion of an electron becomes circular due to the continuous perpendicular force acting on it. The magnetic force does not change the speed of the electron, only the direction, which results in circular motion. This behavior is similar to how a ball tied to a string moves in a circle when swung. In the example, because the magnetic force is directed toward the observer (due to the negative charge), the electron moves in a circular path. The path traces out a plane that is perpendicular to both the direction of the magnetic field and the initial velocity of the electron. Observing from above, this motion appears clockwise.
Uniform Magnetic Field
A uniform magnetic field is characterized by having a constant magnitude and direction throughout a given region. This uniformity ensures that any charged particle, such as an electron moving through the field, experiences a consistent magnetic force. In our situation, the magnetic field is directed horizontally away from the observer, with a strength of \(0.480 \mathrm{~T}\).The uniform nature of the field plays a critical role in maintaining a consistent circular path for the electron. Due to this steady influence, the electron does not experience any changes in speed, and the path remains continuous and predictable. Thus, the uniform magnetic field is essential for the electron鈥檚 circular motion, as it provides the necessary conditions for constant velocity change along a circular trajectory.

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Most popular questions from this chapter

In a probe that uses the Hall effect to measure magnetic fields, a 12.0-A current passes through a 1.50 -cm-wide 1.30-mm-thick strip of sodium metal. If the Hall emf is \(1.86 \mu \mathrm{V},\) what is the magnitude of the magnetic field (take it perpendicular to the flat face of the strip)? Assume one free electron per atom of \(\mathrm{Na}\), and take its specific gravity to be 0.971 .

The Hall effect can be used to measure blood flow rate because the blood contains ions that constitute an electric current. (a) Does the sign of the ions influence the emf? (b) Determine the flow velocity in an artery \(3.3 \mathrm{~mm}\) in diameter if the measured emf is \(0.13 \mathrm{mV}\) and \(B\) is \(0.070 \mathrm{~T}\). (In actual practice, an alternating magnetic field is used.)

(II) A Hall probe, consisting of a rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude 0.10 T. When the field is oriented normal to the slab's rectangular face, a Hall emf of 12 \(\mathrm{mV}\) is measured across the slab's width. The probe is then placed in a magnetic field of unknown magnitude \(B\) , and a Hall emf of 63 \(\mathrm{mV}\) is measured. Determine \(B\) assuming that the angle \(\theta\) between the unknown field and the plane of the slab's rectangular face is \((a) \theta=90^{\circ},\) and \((b) \theta=60^{\circ} .\)

A stiff wire \(50.0 \mathrm{~cm}\) long is bent at a right angle in the middle. One section lies along the \(z\) axis and the other is along the line \(y=2 x\) in the \(x y\) plane. A current of \(20.0 \mathrm{~A}\) flows in the wire \(-\) down the \(z\) axis and out the line in the \(x y\) plane. The wire passes through a uniform magnetic field given by \(\overrightarrow{\mathbf{B}}=(0.318 \hat{\mathbf{i}})\) T. Determine the magnitude and direction of the total force on the wire.

(II) Suppose a straight 1.00 -mm-diameter copper wire could just "float" horizontally in air because of the force due to the Earth's magnetic field \(\vec{\mathbf{B}},\) which is horizontal, perpendic- ular to the wire, and of magnitude \(5.0 \times 10^{-5} \mathrm{T}\) . What current would the wire carry? Does the answer seem feasible? Explain briefly.
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