Problem 89 - College Physics

Question

\(\bullet\) Atom smashers! A cyclotron particle accelerator (sometimes called an "atom smasher" in the popular press) is a device for accelerating charged particles, such as electrons and protons, to speeds close to the speed of light. The basic design is quite simple. The particle is bent in a circular path by a uniform magnetic field. An electric field is pulsed periodically to increase the speed of the particle. The charged particle (or ion) of mass \(m\) and charge \(q\) is introduced into the cyclotron so that it is moving perpendicular to a uniform magnetic field \(\vec{B}\) (a) Starting with the radius of the circular path of a charge moving in a uniform magnetic field, show that the time \(T\) for this particle to make one complete circle is \(T=\frac{2 \pi m}{|q| B}\) . (Hint: You can express the speed \(v\) in terms of \(R\) and \(T\) because the particle travels through one circumference of the circle in time \(T\) . (b) Which would take longer to complete one circle, an ion moving in a large circle or one moving in a small circle? Explain.

Step-by-Step Solution

Answer

(a) Derived formula: \( T = \frac{2\pi m}{|q|B} \). (b) Time is independent of radius, same for any circle size.

Step 1: Understand the Circular Motion

When a charged particle moves in a magnetic field, it follows a circular path. The centripetal force that keeps the particle moving in a circle is provided by the magnetic force: \( F_B = qvB \).

Step 2: Equate Forces

In circular motion, the centripetal force is expressed as \( F_c = \frac{mv^2}{R} \), where \( R \) is the radius of the path. Set \( F_B = F_c \) to get \( qvB = \frac{mv^2}{R} \).

Step 3: Solve for Velocity

From \( qvB = \frac{mv^2}{R} \), solve for \( v \): \( v = \frac{qBR}{m} \).

Step 4: Relate Velocity to Time Period

The velocity \( v \) is also the circumference divided by time, \( v = \frac{2\pi R}{T} \). Substitute \( v = \frac{qBR}{m} \) into this equation to get \( \frac{qBR}{m} = \frac{2\pi R}{T} \).

Step 5: Solve for Time Period

Cancel \( R \) from both sides of \( \frac{qBR}{m} = \frac{2\pi R}{T} \) and solve for \( T \): \( T = \frac{2\pi m}{|q|B} \).

Step 6: Comparison of Circle Sizes

The time \( T \) is independent of the radius \( R \); it only depends on the mass \( m \), charge \( q \), and magnetic field \( B \). Both ions, regardless of their path size, take the same time to complete one circle assuming constant \( m \), \( q \), and \( B \).

Key Concepts

Charged Particles

Charged particles are essential components in the realm of physics, especially when it comes to devices like cyclotron particle accelerators. These particles, such as electrons and protons, possess an electric charge. This charge is crucial because it allows them to interact with electric and magnetic fields. When a charged particle enters a magnetic field, it experiences a magnetic force which can alter its path.

In the context of a cyclotron, these charged particles are accelerated to high speeds using these fields. The electric field continually nudges the particle's energy, increasing its speed, while the magnetic field guides it into a circular orbit. Some key points to remember about charged particles are:

• They have either a positive or negative charge.
• Their interactions with fields depend on their charge and velocity.
• In accelerators, their mass and charge determine their acceleration and path.

Magnetic Field

A magnetic field is a region around magnetic materials or moving electric charges where the force of magnetism acts. In a cyclotron, a uniform magnetic field is used, meaning the field has the same strength and direction throughout the space it occupies. This uniformity is vital for ensuring that particles experience a consistent force, guiding them smoothly along a circular path.

The force exerted by the magnetic field on a charged particle is given by the equation:

• oindent \( F_B = qvB \)

This equation tells us that the magnetic force \( F_B \) is directly proportional to the charge \( q \), velocity \( v \), and magnetic field strength \( B \). The direction of the force is always perpendicular to both the magnetic field and the direction of motion. This causes the particle to move in a circle, not speeding up or slowing down, but merely changing direction.

Circular Motion

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. When it comes to charged particles in a cyclotron, this circular motion is crucial. The particle moves in a constant circle due to the centripetal force provided by the magnetic field.

• The centripetal force required for this motion keeps changing the particle's direction, making it follow a circular path.
• The force doesn’t do work on the particle, so its speed remains consistent unless energy is added via the electric field.
• For a particle of mass \( m \) traveling with speed \( v \) and radius \( R \), the centripetal force \( F_c \) is expressed as: \[ F_c = \frac{mv^2}{R} \].

By maintaining this balance of forces, the particle continues its motion in a circle, a central aspect of operation in particle accelerators.

Centripetal Force

Centripetal force is the required force to make a body follow a curved path. In the case of charged particles in a cyclotron, this force is provided entirely by the magnetic force exerted on the particle. As we established, the magnetic force \( F_B \) serves as this centripetal force, continuously acting perpendicular to the particle's velocity, bending its path without altering its speed.

• This is well represented by the formula: \[ F_c = \frac{mv^2}{R} = qvB \].
• It ensures that the particle does not fly off in a straight line. Instead, it maintains its circular trajectory within the cyclotron.
• The magnitude of the centripetal force depends on the particle’s mass and velocity for a given circle.

Understanding how centripetal force works in tandem with magnetic forces is critical for comprehending how cyclotrons function to accelerate particles.