Problem 56 - University Physics with Modern Physics

Question

A cycloron is to accelerate protons to an energy of 5.4 MeV. The superconducting electromagnet of the cyclotron produces a \(3.5-\) T magnetic field perpendicular to the proton orbits. (a) When the protons have achieved a kinetic energy of 2.7 \(\mathrm{MeV}\) , what is the radius of their circular orbit and what is their angular speed? (b) Repeat part (a) when the protons have achieved their final kinetic energy of 5.4 \(\mathrm{MeV}\) .

Step-by-Step Solution

Answer

At 2.7 MeV, radius ≈ 0.095 m, angular speed ≈ 2.1 × 10⁸ rad/s. At 5.4 MeV, radius ≈ 0.135 m, angular speed ≈ 1.5 × 10⁸ rad/s.

Step 1: Understanding the Problem

We need to find the radius of the circular orbit and the angular speed of protons when they have kinetic energies of 2.7 MeV and 5.4 MeV in a cyclotron with a magnetic field of 3.5 T.

Step 2: Recall Relevant Formulas

Use the formula for the radius of a charged particle moving in a magnetic field, given by \( r = \frac{mv}{qB} \), where \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field. The angular speed \( \omega \) is given by \( \omega = \frac{v}{r} \). The kinetic energy can be related to velocity by \( KE = \frac{1}{2}mv^2 \).

Step 3: Calculate Velocity from Kinetic Energy for 2.7 MeV

The kinetic energy \( KE = 2.7 \, \text{MeV} = 2.7 \times 10^6 \times 1.6 \times 10^{-19} \, \text{J} \). Well-known constants are the proton mass \( m = 1.67 \times 10^{-27} \, \text{kg} \) and charge \( q = 1.6 \times 10^{-19} \, \text{C} \). Calculate the velocity using \( v = \sqrt{\frac{2KE}{m}} \).

Step 4: Compute Radius for 2.7 MeV

Calculate the radius \( r \) of the orbit using \( r = \frac{mv}{qB} \), substituting the values found for \( m, v, q, \) and \( B = 3.5 \, \text{T} \).

Step 5: Find Angular Speed for 2.7 MeV

Calculate the angular speed \( \omega \) using \( \omega = \frac{v}{r} \) with the previously computed \( v \) and \( r \).

Step 6: Repeat Calculations for 5.4 MeV

Use the same process in Steps 3 to 5 but substitute \( KE = 5.4 \, \text{MeV} \) to find the radius and angular speed for protons at this energy.

Key Concepts

Proton Acceleration

A cyclotron is a type of particle accelerator that is used to speed up charged particles, like protons, to high energies. When working with a cyclotron, one of the main goals is to increase the kinetic energy of these protons so that they move faster and with more force. This acceleration process is achieved using both electric and magnetic fields.

Electric fields provide the push that protons need to get moving. Every time the proton crosses the gap between the D-shaped electrodes (commonly known as dees), it gains energy from the electric field. This energy boosts the speed of the proton, incrementally increasing its kinetic energy. Over several cycles within the cyclotron chamber, this continuous increase in speed helps the proton achieve a desired energy level, such as 2.7 MeV or 5.4 MeV.

In this cyclotron exercise, the role of acceleration is crucial, as we track how kinetic energy changes the motion characteristics of protons. To achieve such high energy, careful engineering and precise control of the electric fields are necessary to ensure the protons follow the intended path and gain energy efficiently.

Magnetic Field

The magnetic field in a cyclotron plays a crucial role in directing and controlling the path of protons. As the protons gain energy and speed through the cyclotron's electric field, the magnetic field ensures that they move in circular paths inside the chamber.

In this scenario, the cyclotron uses a magnetic field of 3.5 Tesla, which is a powerful force. This magnetic field acts perpendicular to the path of the protons. According to the principles of electromagnetism, when a charged particle, such as a proton, enters a magnetic field perpendicularly, it experiences a force that causes it to move in a circular trajectory. This force is known as the Lorentz force.

The strength of the magnetic field determines how tight or wide the proton's circular path will be. A stronger magnetic field will make the circle smaller, while a weaker field allows for a larger orbit. This characteristic of the magnetic field is essential for calculating the radius of the orbit and understanding how it affects the proton's motion as it gains energy.

Circular Orbit Radius

The radius of the circular orbit in a cyclotron is a key metric that describes the path of a proton as it travels within the magnetic field. This radius depends on various parameters, including the speed of the proton, the magnetic field strength, and the specific properties of the proton itself, such as its mass and charge.

In the formula for calculating the radius of the circular orbit, expressed as \( r = \frac{mv}{qB} \), each element plays a vital role:

• \( m \) is the mass of the proton.
• \( v \) is the velocity of the proton.
• \( q \) is the charge of the proton.
• \( B \) is the magnetic field's strength.

As the kinetic energy of the proton increases, its velocity \( v \) does too, causing changes in the orbit radius. For example, at 2.7 MeV, the orbit radius will have a certain value. At a higher energy level of 5.4 MeV, this radius will increase, reflecting the increase in the proton's speed. Understanding this dynamic relationship helps in designing cyclotrons and predicting proton behavior within the magnetic field.

Angular Speed

Angular speed \( \omega \) is a measure of how fast an object is rotating or moving along a circular path. In the context of a cyclotron, understanding the angular speed of protons gives insight into their dynamic behavior as they move within the magnetic field.

To calculate angular speed, we use the formula \( \omega = \frac{v}{r} \), where \( v \) is the velocity of the proton, and \( r \) is the radius of its circular path. Angular speed increases as protons gain more kinetic energy. For instance, as a proton's energy changes from 2.7 MeV to 5.4 MeV, its velocity increases, which results in a change in angular speed.

Angular speed is particularly important in determining the frequency of the proton's rotation within the cyclotron. This frequency is crucial for ensuring that proton acceleration is synchronized with the oscillating electric field, allowing for efficient energy transfer and a stable increase in speed and energy.