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Chapter 20: Problem 17

A uniform magnetic field bends an electron in a circular arc of radius \(R .\) What will be the radius of the arc (in terms of \(R )\) if the field is tripled?

Short Answer

Expert verified
The new radius is \( \frac{R}{3} \).

Step by step solution

01

Understanding the Problem

We are asked to find the new radius of a circular arc when the magnetic field acting on an electron is tripled. We start by understanding the motion of an electron in a magnetic field, which moves in a circular path due to the Lorentz force.
02

Using the Lorentz Force

The Lorentz force, which causes the electron to move in a circular path, is given by \( F = qvB \), where \( q \) is the charge of the electron, \( v \) is its velocity, and \( B \) is the magnetic field strength.
03

Centripetal Force Equation

For circular motion, the centripetal force required can be expressed as \( F = \frac{mv^2}{R} \), where \( m \) is the mass of the electron and \( R \) is the radius of the circle.
04

Equating Forces

Setting the centripetal force equal to the Lorentz force, we have \( qvB = \frac{mv^2}{R} \). From this, we can solve for \( R \): \[ R = \frac{mv}{qB} \].
05

Finding the New Radius

If the magnetic field is tripled, the new field strength is \( 3B \). Plugging this into our equation for \( R \), we have: \[ R' = \frac{mv}{q(3B)} = \frac{1}{3} \times \frac{mv}{qB} = \frac{1}{3}R \].
06

Conclusion

The new radius of the arc, when the magnetic field is tripled, becomes one-third of the original radius \( R \).

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Key Concepts

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

Lorentz Force
When an electron moves through a magnetic field, it experiences a force known as the Lorentz force. This force is responsible for the curved path an electron follows in such a field. The force is a result of the interaction between the magnetic field and the moving charge of the electron. It can be calculated using the formula \( F = qvB \). Here, \( q \) represents the charge of the electron (approximately \(-1.6 \times 10^{-19} \) coulombs), \( v \) is the velocity of the electron, and \( B \) is the magnetic field strength.

Key points about Lorentz force:
  • The direction of the Lorentz force is perpendicular to both the velocity of the electron and the magnetic field. This is what causes the charge to move in a circular manner.
  • This force alters only the direction of velocity, changing kinetic energy or speed of the particle.
  • Understanding how Lorentz force operates helps in applications like mass spectrometry and particle accelerators.
Circular Motion
When a force acts perpendicular to the velocity of an object, like Lorentz force on an electron, it induces circular motion. In a magnetic field, electrons naturally proceed in a circular path because the force constantly changes their direction rather than their speed. This phenomenon is fundamental in physics, describing how charges like electrons behave under magnetic influences.

Features of circular motion due to a magnetic field:
  • The radius of the motion can be described by the equation \( R = \frac{mv}{qB} \), indicating that the velocity and magnetic field strength greatly affect the path the electron traces.
  • If the magnetic field is stronger, the curvature becomes tighter, reducing the radius.
  • Accurate calculation of motion parameters like radius helps in designing devices that use magnetic fields to control charged particles.
Centripetal Force
In circular motion, centripetal force is the inward force that maintains an object's curved path. For electrons moving in magnetic fields, the Lorentz force serves as this centripetal force. It's the necessary pull that prevents the electron from flying off in a straight line.

Centripetal force characteristics:
  • It can be expressed as \( F = \frac{mv^2}{R} \), where \( m \) is the mass, \( v \) is velocity, and \( R \) is the radius.
  • By equating this expression with the Lorentz force, \( F = qvB \), we derive the formula for the radius \( R \) and understand how the electron's path adjusts with changes in magnetic field.
  • The concept is widely applicable in technologies utilizing rotating systems, ensuring stable motion.

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Most popular questions from this chapter

\(\bullet\) A beam of protons is accelerated through a potential dif- ference of 0.745 \(\mathrm{kV}\) and then enters a uniform magnetic field traveling perpendicular to the field. (a) What magnitude of field is needed to bend these protons in a circular arc of diameter 1.75 \(\mathrm{m} ?\) (b) What magnetic field would be needed to produce a path with the same diameter if the particles were electrons having the same speed as the protons?

Two long parallel transmission lines 40.0 \(\mathrm{cm}\) apart carry 25.0 \(\mathrm{A}\) and 75.0 A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction, (b) opposite directions.

Household magnetic fields. Home circuit breakers typically have current capacities of around 10 A. How large a magnetic field would such a current produce 5.0 \(\mathrm{cm}\) from a long- wire's center? How does this field compare with the strength of the earth's magnetic field?

(a) What is the speed of a beam of electrons when the simultaneous influence of an electric field of \(1.56 \times 10^{4} \mathrm{V} / \mathrm{m}\) and a magnetic field of \(4.62 \times 10^{-3} \mathrm{T}\) , with both fields normal to the beam and to each other, produces no deflection of the electrons? (b) In a diagram, show the relative orientation of the vectors \(\vec{\boldsymbol{v}}\) , \(\vec{\boldsymbol{E}}\) and \(\vec{\boldsymbol{B}}\) . (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of \(6.64 \times 10^{-27}\) kg traveling horizontally at 35.6 \(\mathrm{km} / \mathrm{s}\) enters a uniform, vertical, 1.10 \(\mathrm{T}\) magnetic field. (a) What is the diameter of the path followed by this alpha particle? (b) What effect does the magnetic field have on the speed of the particle? (c) What are the magnitude and direction of the acceleration of the alpha particle while it is in the magnetic field? (d) Explain why the speed of the particle does not change even though an unbalanced external force acts on it.
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