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Chapter 27: Problem 14

(I) 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 will move in a counterclockwise circular path in the vertical plane away from the observer.

Step by step solution

01

Understanding the Problem

An electron is moving vertically upward into a magnetic field. The magnetic field is horizontal and directed away from the observer. We want to describe the path of the electron in this magnetic field.
02

Determining the Direction of Force

The direction of the force experienced by a charged particle due to a magnetic field is given by the right-hand rule. However, because the electron is negatively charged, we reverse the direction given by the right-hand rule. Place your thumb in the direction of velocity (upwards), your fingers in the direction of the magnetic field (away), and the force acting on the electron will be perpendicular to both, pointing to the left.
03

Analyzing Motion

The force acting on the electron will cause it to move in a circular path because the magnetic force is always perpendicular to the velocity of the electron. This results in a centripetal force that causes uniform circular motion.
04

Determining Path Direction

Since the magnetic field is directed horizontally and the velocity is vertically upward, the motion of the electron will be a circle in a plane perpendicular to the magnetic field, i.e., in the vertical plane. The path will be circular and, considering the initial leftward force, it will be counterclockwise when viewed from the perspective of the observer.

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Key Concepts

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

Electron Motion in Magnetic Field
When an electron enters a magnetic field, it experiences a force due to its charge and velocity in the field. This force changes the particle's path from a straight line to a curved one.
In the given scenario, the electron is moving vertically upward as it enters a horizontal magnetic field which points away from the observer. Despite initially traveling straight, this force caused by the magnetic field will curve its path.
The magnetic force will act in a direction perpendicular to the velocity of the electron and the field lines.
  • The electron's charge being negative means it will experience a force in the opposite direction predicted by the right-hand rule.
  • The force will cause the electron to deviate from its initial path, eventually moving in a circular trajectory.
As the magnetic force is always perpendicular to the velocity, it changes only the direction of the electron's motion, not its speed.
Right-Hand Rule
To determine the direction of magnetic force experienced by charged particles like electrons, we use the right-hand rule. Typically, you'd use your right hand but need to reverse the outcome for negatively charged particles like electrons.
Here's how you can apply it in this context:
  • Align your thumb with the exact direction of the velocity of the particle. Since the electron is moving vertically upwards, point your thumb upwards.
  • Next, extend your fingers in the direction of the magnetic field. Here, the field is horizontal and directed away from you, so point your fingers in that direction.
  • Your palm will naturally face in the direction of the force on a positive charge. Yet, electrons have a negative charge, so the real force direction is opposite to the natural palm direction.
In this case, the force will direct to the left of the electron. This reversal is exclusive to negatively charged particles like electrons.
Uniform Circular Motion
Uniform circular motion occurs when an object moves along a circular path at constant speed. In the context of an electron in a magnetic field, this movement is due to the centripetal force provided by the magnetic force.
Since the magnetic force is always perpendicular, it acts as a centripetal force resulting in circular motion.
  • Even though the speed remains constant, the direction of the velocity constantly changes, creating the circular path.
  • This centripetal force constrains the electron to continue moving in this circular manner until it exits the magnetic field or other forces act upon it.
It's important to note that the radius of this circle is determined by factors such as the speed of the electron and the strength of the magnetic field.
In our example, given the combined left-hand force and the initial upward velocity, the path pursued by the electron resembles a counterclockwise circle when viewed from the observer's viewpoint.

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Most popular questions from this chapter

Two stiff parallel wires a distance \(d\) apart in a horizontal plane act as rails to support a light metal rod of mass \(m\) (perpendicular to each rail), Fig. \(27-49 .\) A magnetic field \(\overrightarrow{\mathbf{B}}\) directed vertically upward (outward in diagram), acts throughout. At \(t=0\), a constant current \(I\) begins to flow through the system. Determine the speed of the rod, which starts from rest at \(t=0\), as a function of time \((a)\) assuming no friction between the rod and the rails, and \((b)\) if the coefficient of friction is \(\mu_{\mathrm{k}} .(c)\) In which direction does the rod move, east or west, if the current through it heads north?

(II) Show that the magnetic dipole moment \(\mu\) of an electron orbiting the proton nucleus of a hydrogen atom is related to the orbital momentum \(L\) of the electron by $$ \mu=\frac{e}{2 m} L $$

In a certain cathode ray tube, electrons are accelerated horizontally by \(25 \mathrm{kV}\). They then pass through a uniform magnetic field \(B\) for a distance of \(3.5 \mathrm{~cm},\) which deflects them upward so they reach the top of the screen \(22 \mathrm{~cm}\) away, \(11 \mathrm{~cm}\) above the center. Estimate the value of \(B\).

In a mass spectrometer, germanium atoms have radii of curvature equal to \(21.0,21.6,21.9,22.2,\) and \(22.8 \mathrm{~cm} .\) The largest radius corresponds to an atomic mass of \(76 \mathrm{u}\). What are the atomic masses of the other isotopes?

(I) If the restoring spring of a galvanometer weakens by 15\(\%\) over the years, what current will give full-scale deflection if it originally required 46\(\mu \mathrm{A}\) ?
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