Question

A cyclotron in a magnetic field of \(9.00 \mathrm{~T}\) is used to accelerate protons to \(50.0 \%\) of the speed of light. What is the cyclotron frequency of these protons? What is the radius of their trajectory in the cyclotron? What are the cyclotron frequency and the trajectory radius of the same protons in the Earth's magnetic field? Assume that the Earth's magnetic field is 0.500 G.

Short answer

Question: Calculate the cyclotron frequencies and trajectory radius for protons in a 9.00 T magnetic field and Earth's magnetic field (0.500 G), given that the protons will reach 50.0% of the speed of light. Answer: For the 9.00 T magnetic field: Cyclotron frequency: \(f = 7.63 \times 10^7\,\text{Hz}\) Trajectory radius: \(r = 0.233\,\text{m}\) For Earth's magnetic field (0.500 G): Cyclotron frequency: \(f_{earth} = 3.82 \times 10^3\,\text{Hz}\) Trajectory radius: \(r_{earth} = 4.65 \times 10^5\,\text{m}\)

Step-by-step solution

Calculate the cyclotron frequency in the given magnetic field.

Using the cyclotron frequency formula mentioned above, we can calculate the frequency for the protons in the 9.00 T magnetic field: $$ f = \frac{qB}{2\pi m} $$ Given the charge of a proton is \(q = 1.602 \times 10^{-19}\,\text{C}\) and mass of a proton is \(m = 1.673 \times 10^{-27}\,\text{kg}\), we can plug in the values for \(q\), \(B\), and \(m\): $$ f = \frac{(1.602 \times 10^{-19}\,\text{C})(9.00\,\text{T})}{2\pi(1.673 \times 10^{-27}\,\text{kg})} $$

Calculate the radius of the trajectory in the given magnetic field.

Since the protons are going to reach \(50.0 \%\) of the speed of light, their final speed will be $$ v = 0.50 \times c = 0.50 (3.00 \times 10^8\,\text{m/s}) $$ Now, we can use the trajectory radius formula to find the radius: $$ r = \frac{mv}{qB} $$ Plug in the values for \(m\), \(v\), \(q\), and \(B\): $$ r = \frac{(1.673 \times 10^{-27}\,\text{kg})(0.50 (3.00 \times 10^8\,\text{m/s}))}{(1.602 \times 10^{-19}\,\text{C})(9.00\,\text{T})} $$ Now, let's find the cyclotron frequency and radius of trajectory for the same protons in the Earth's magnetic field (0.500 G). Note that, 1 tesla = \(10^4\) G, so \(0.500\,\text{G}\) = \(5.00 \times 10^{-5}\,\text{T}\).

Calculate the cyclotron frequency in the Earth's magnetic field.

To find the cyclotron frequency in Earth's magnetic field, we use the same formula as before, but with the Earth's magnetic field value: $$ f_{earth} = \frac{qB_{earth}}{2\pi m} $$ Plug in the values for \(q\), \(B_{earth}\), and \(m\): $$ f_{earth} = \frac{(1.602 \times 10^{-19}\,\text{C})(5.00 \times 10^{-5}\,\text{T})}{2\pi(1.673 \times 10^{-27}\,\text{kg})} $$

Calculate the radius of trajectory in the Earth's magnetic field.

We can find the radius of trajectory in Earth's magnetic field using the same trajectory radius formula as before: $$ r_{earth} = \frac{mv}{qB_{earth}} $$ Plug in the values for \(m\), \(v\), \(q\), and \(B_{earth}\): $$ r_{earth} = \frac{(1.673 \times 10^{-27}\,\text{kg})(0.50 (3.00 \times 10^8\,\text{m/s}))}{(1.602 \times 10^{-19}\,\text{C})(5.00 \times 10^{-5}\,\text{T})} $$ Finally, evaluate the numerical values for the cyclotron frequencies and trajectory radius in both magnetic field cases.

Key concepts

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A cyclotron utilizes a powerful magnetic field to accelerate charged particles, such as protons. The strength of this field, measured in teslas (T) or gauss (G), massively influences the behavior of particles within it.

When a charged particle moves through a magnetic field, it experiences a force that acts perpendicular to both the direction of its velocity and the magnetic field. This force does not do work on the particle because it always acts perpendicular to the velocity, meaning the particle's speed remains constant, but its direction changes. This phenomenon is fundamental in understanding how a cyclotron works, as the magnetic field forces the particles into a spiral path, continually accelerating them until they reach the desired speed.

In our exercise, a significant 9.00 T magnetic field accelerates protons to high speeds. Interestingly, when comparing this to the Earth's magnetic field, which is about 0.500 G (or 0.00005 T), it's clear why cyclotrons need such strong magnetic fields to function effectively. Protons would spiral much further and take longer to accelerate in Earth's relatively weak magnetic field.

Trajectory Radius

The trajectory radius in a cyclotron refers to the spiral path's radius that a charged particle follows while being accelerated by the magnetic field. According to physics, the radius of this trajectory can be calculated using the formula:
\r\[\begin{equation}\rr = \frac{mv}{qB}\r\end{equation}\]\r
Here, 'm' represents the particle's mass, 'v' is its velocity, 'q' is the charge, and 'B' is the magnetic field's strength. For the protons in our exercise, as they reach 50% of the speed of light, their large velocities combined with the effects of the strong magnetic field create a relatively compact trajectory radius within the cyclotron.

This radius is crucial because it also indicates the size of the cyclotron needed for a given particle acceleration task. Higher velocities or stronger magnetic fields would result in smaller trajectory radii. This concept is directly linked to cyclotron frequency, as a shorter path requires a higher frequency to complete each circle in the same period.

Proton Acceleration

Proton acceleration in a cyclotron occurs due to the presence of both a magnetic field and an alternating electric field. The magnetic field keeps protons moving in a circular path while the electric field is used to speed them up every time they cross the gap between the cyclotron's 'D-shaped' electrodes, known as dees.

To determine how quickly the protons gain speed, the cyclotron frequency is used and given by:\r\[\begin{equation}\rf = \frac{qB}{2\pi m}\r\end{equation}\]\r
Identifying this frequency is essential because it must match the frequency of the alternating electric field to successfully accelerate the protons. If not matched, the protons could decelerate. The high magnetic field of 9.00 T in a cyclotron allows for the protons to reach up to 50% of the speed of light, demonstrating a significant amount of energy gained during the acceleration process.

In summary, both cyclotron frequency and trajectory radius are core aspects of proton acceleration within a cyclotron. Understanding their correlation and how they change with different magnetic field strengths can give us insights into the principles governing cyclotrons and particle accelerators at large.