Problem 57 - University Physics with Modern Physics

Question

The magnetic poles of a small cyclotron produce a magnetic field with magnitude 0.85 \(\mathrm{T}\) . The poles have a radius of \(0.40 \mathrm{m},\) which is the maximum radius of the orbits of the accelerated particles. (a) What is the maximum energy to which protons \(\left(q=1.60 \times 10^{-19} \mathrm{C}, m=1.67 \times 10^{-27} \mathrm{kg}\right)\) can be accelerated by this cyclotron? Give your answer in electron volts and in joules. (b) What is the time for one revolution of a proton orbiting at this maximum radius? (c) What would the magnetic-field magnitude have to be for the maximum energy to which a proton can be accelerated to be twice that calculated in part (a)? (d) For \(B=0.85 \mathrm{T}\) . what is the maximum energy to which alpha particles \(\left(q=3.20 \times 10^{-19} \mathrm{C}, m=6.65 \times 10^{-27} \mathrm{kg}\right)\) can be accelerated by this cyclotron? How does this compare to the maximum energy for protons?

Step-by-Step Solution

Answer

Protons gain max energy 5.53 MeV; fastest cycle time is 49.2 ns; double energy needs 1.20 T; alphas gain 11.1 MeV.

Step 1: Understand the Problem

We need to find the maximum kinetic energy protons can achieve in a cyclotron with a given magnetic field and radius. Additionally, we calculate the revolution time for these protons, determine the magnetic field for double the energy, and find energies for alpha particles.

Step 2: Calculate Maximum Velocity using the Cyclotron Equation

The velocity of protons in a cyclotron is determined by the radius and the magnetic field.\[ v = \frac{qBr}{m} \]Substitute the given values:\[ v = \frac{(1.60 \times 10^{-19} \, \text{C})(0.85 \, \text{T})(0.40 \, \text{m})}{1.67 \times 10^{-27} \, \text{kg}} = 3.25 \times 10^7 \, \text{m/s} \]

Step 3: Calculate Maximum Kinetic Energy

The kinetic energy is given by:\[ K.E. = \frac{1}{2}mv^2 \]Substitute the values found in the previous step:\[ K.E. = \frac{1}{2}(1.67 \times 10^{-27} \, \text{kg})(3.25 \times 10^7 \, \text{m/s})^2 = 8.85 \times 10^{-13} \,\text{J} \]Convert joules to electron volts:\[ K.E. = \frac{8.85 \times 10^{-13} \,\text{J}}{1.60 \times 10^{-19} \,\text{J/eV}} = 5.53 \times 10^6 \, \text{eV} \]

Step 4: Calculate Revolution Time for Protons

The time for one revolution in a cyclotron is:\[ T = \frac{2\pi m}{qB} \]Substitute the known values:\[ T = \frac{2\pi (1.67 \times 10^{-27} \, \text{kg})}{(1.60 \times 10^{-19} \, \text{C})(0.85 \, \text{T})} = 4.92 \times 10^{-8} \, \text{s} \]

Step 5: Modify Magnetic Field for Double Energy

If the energy needs to be doubled, the speed will increase by a factor of \(\sqrt{2}\), requiring the magnetic field to also increase:\[ B' = \sqrt{2} B = \sqrt{2}(0.85 \, \text{T}) = 1.20 \, \text{T} \]

Step 6: Calculate Maximum Energy for Alpha Particles

Using the same formula as for protons, find the energy with different charge and mass values:\[ K.E._{\alpha} = \frac{1}{2}(6.65 \times 10^{-27} \, \text{kg})\left(\frac{(3.20 \times 10^{-19} \, \text{C})(0.85 \, \text{T})(0.40 \, \text{m})}{6.65 \times 10^{-27} \, \text{kg}}\right)^2 = 1.77 \times 10^{-12} \, \text{J} \]Convert to electron volts:\[ K.E._{\alpha} = \frac{1.77 \times 10^{-12} \,\text{J}}{1.60 \times 10^{-19} \,\text{J/eV}} = 1.11 \times 10^7 \, \text{eV} \]

Step 7: Comparison of Energies

The maximum energy for alpha particles (\(1.11 \times 10^7 \, \text{eV}\)) is about twice the energy for protons (\(5.53 \times 10^6 \, \text{eV}\)) due to the greater charge and mass in alpha particles.

Key Concepts

Proton acceleration

In a cyclotron, particles like protons are accelerated using a high magnetic field and radio frequency electric fields. The cyclotron propels protons in a circular path, gaining speed with each revolution. It achieves this through a combination of electric fields that accelerate the protons and a magnetic field that bends their paths into a circle. The kinetic energy obtained by a proton as it travels is linked to the speed it reaches by the equation:

• \[ v = \frac{qBr}{m} \]

Where:

• \( v \) is the velocity;
• \( q \) is the charge of the particle;
• \( B \) is the magnetic field strength;
• \( r \) is the radius of the cyclotron's magnetic field area;
• \( m \) is the mass of the proton.

This velocity is crucial for determining the kinetic energy, which directly impacts the applications where these protons can be used.

Magnetic field strength

The magnetic field strength in a cyclotron, measured in Tesla (T), is pivotal in determining the particle trajectory and velocity. A stronger magnetic field enables greater bending of the charged particle's path, which is vital for maintaining its circular motion. The relationship between the magnetic field and the particle velocity is direct. With the formula:

• \[ B = \frac{mv}{qr} \]

This equation helps us understand how the magnetic field strength can influence the overall acceleration process, enabling precise control over the particle's speed and orbit size. Strengthening the magnetic field for a specific frequency ensures that the protons are effectively synchronized with the radio-frequency electric field.

Maximum kinetic energy

The kinetic energy of particles such as protons or alpha particles in a cyclotron is a measure of their motion energy, influenced by their velocity and mass. Kinetic energy \( K.E. \) is calculated using the formula:

• \[ K.E. = \frac{1}{2}mv^2 \]

The maximum kinetic energy achieved by a proton depends on its charging path (electric field) and bending path (magnetic field). This energy is essential for experiments like nuclear reactions as it indicates the particle's capability to overcome barriers such as those between atomic nuclei. The cyclotron must balance between the technical potential of the accelerating system and safety standards, as high kinetic energies can be hazardous.

Alpha particles

Alpha particles are helium nuclei, consisting of two protons and two neutrons, giving them a higher mass and charge than single protons. When accelerated in a cyclotron, their behavior is similar but distinct due to the greater mass and double charge:

• Charge \( q = 3.20 \times 10^{-19} \, \text{C} \)
• Mass \( m = 6.65 \times 10^{-27} \, \text{kg} \)

These differences result in different paths and maximum kinetic energy levels compared to protons. Although they achieve higher energy levels \( 1.11 \times 10^7 \, \text{eV} \), the increase in mass means they require more energy to reach a similar velocity, creating interesting dynamics in comparative experiments.

Cyclotron frequency

Cyclotron frequency refers to the rate at which a charged particle orbits inside the device. It is essential to synchronize this frequency with the alternating voltage applied across the cyclotron's gaps, maximizing particle acceleration. The frequency is given by:

• \[ f = \frac{qB}{2\pi m} \]

This ensures that the electric field accelerates the particles at the precise moments they cross the gaps. Synchronizing this frequency with the high-frequency voltage increases the energy particles can gain per cycle, making the cyclotron efficient. Understanding and controlling cyclotron frequency allows for optimal operation, helping scientists achieve desired energy and stability levels for experimental and medical applications.