Question

The magnetic field in a cyclotron that accelerates protons is 1.70 T. (a) How many times per second should the potential across the dees reverse? (This is twice the frequency of the circulating protons.) (b) The maximum radius of the cyclotron is 0.250 m. What is the maximum speed of the proton? (c) Through what potential difference must the proton be accelerated from rest to give it the speed that you calculated in part (b)?

Short answer

(a) The potential reverses at 25.8 MHz. (b) The maximum speed is \( 4.08 \times 10^7 \) m/s. (c) The required potential difference is 438 kV.

Step-by-step solution

Understanding the Cyclotron Frequency

The cyclotron frequency is the rate at which the proton circulates in the magnetic field, and it is given by the formula \(f = \frac{qB}{2\pi m}\), where \(q\) is the charge of the proton, \(B\) is the magnetic field strength (1.70 T here), and \(m\) is the mass of the proton (\(1.67 \times 10^{-27}\) kg). Calculating this gives us the frequency of the proton.

Calculating the Required Frequency of Potentials

The frequency at which the potential across the dees should reverse is twice the cyclotron frequency, \(f' = 2f\). Substitute the value of \(f\) from Step 1 to find \(f'\), which gives the number of times the potential should reverse per second.

Calculating the Maximum Speed of the Proton

The maximum speed of the proton in the cyclotron can be given by \(v_{max} = r_c \cdot q \cdot B / m\), where \(r_c\) is the maximum radius of the cyclotron (0.250 m). Substitute the known values into this equation to find the maximum speed \(v_{max}\).

Determining the Potential Difference for Maximum Speed

Using the energy principle, the kinetic energy \( \frac{1}{2}mv_{max}^2 \) is equal to the energy gained by a proton in an electric field, \(q \cdot V\). Solve for \(V\) by substituting \(v_{max}\) from Step 3.

Key concepts

Magnetic Field Strength

Magnetic field strength is a crucial concept in understanding cyclotron operations.
In the context of a cyclotron, which is a type of particle accelerator, this magnetic field is uniform and plays a significant role in bending charged particles like protons into a circular path. In our example, the magnetic field strength is given as 1.70 Tesla (T). This value indicates the intensity of the magnetic field applied within the cyclotron.The strength of the magnetic field directly affects the cyclotron frequency, as the frequency at which protons circulate depends on this parameter. It is calculated using the formula \[f = \frac{qB}{2\pi m}\]Here, \(q\) represents the charge of the proton, \(B\) is the magnetic field strength, and \(m\) is the mass of the proton.
The magnetic field strength dictates how quickly a charged particle can be looped around its trajectory, thus affecting its acceleration and eventual speed.

Proton Acceleration

Proton acceleration in a cyclotron involves increasing the velocity of protons by applying an oscillating electric field.
Protons gain energy every time they cross the gap between the dees, which are the D-shaped electrodes inside the cyclotron.

• The changing electric potential across these dees accelerates the proton each time it circles through them.
• The reversal of the potential must occur at a frequency that matches twice the cyclotron frequency to keep the protons in sync with the electric field.

This synchronization ensures that protons accumulate energy over multiple circles around the cyclotron without losing pace with the oscillating electric field.
Thus, proton acceleration is a finely tuned process that leverages magnetic fields and precise timing to achieve maximum speed efficiently.

Cyclotron Motion

Cyclotron motion refers to the circular path that charged particles, like protons, follow within the cyclotron. While in a magnetic field, these particles experience a force perpendicular to their movement called the Lorentz force, causing them to move in a circular trajectory.
In cyclotron motion, several factors are crucial:

• The magnetic field causes the circular motion by exerting a constant force.
• The radius of the particle's path, which is directly proportional to its speed and inversely proportional to the magnetic field’s strength. This is given by \[r = \frac{mv}{qB}\]where \(r\) is the radius, \(v\) is the speed, \(m\) and \(q\) are the mass and charge of the proton, respectively.

The cyclotron maintains the protons in this synchronized motion until they reach the perimeter and achieve the desired energy level. This principle is essential to achieving high speeds, as particles successfully gain speed through successive orbits within the cyclotron's confines.

Maximum Speed Calculation

Calculating the maximum speed a proton can achieve in a cyclotron requires understanding the relationship between its physical parameters and energy principles.The maximum speed \(v_{max}\) of a proton can be derived from the equation\[v_{max} = \frac{r_c \cdot q \cdot B}{m}\]where

• \(r_c\) is the maximum radius (0.250 m).
• \(q\) is the charge of the proton.
• \(B\) is the magnetic field strength (1.70 T).
• \(m\) is the mass of the proton.

This expression reflects how the proton's speed depends on the interplay between the cyclotron’s magnetic field and its radius. Once the maximum speed is reached, one can calculate the potential difference required to achieve this energy level using the equation for kinetic energy and electric potential: \[\frac{1}{2}mv_{max}^2 = q \cdot V\]By solving for the potential \(V\), we can find the energy input needed to accelerate the proton to its given speed. This aids in informing design aspects of particle accelerators to achieve desired particle energies.