Question

A cyclotron with dee radius 53.0 cm is operated at an oscillator frequency of 12.0 MHz to accelerate protons.

(a) What magnitude Bof magnetic field is required to achieve resonance?

(b) At that field magnitude, what is the kinetic energy of a proton emerging from the cyclotron? Suppose, instead, that B = 1.57T.

(c) What oscillator frequency is required to achieve resonance now?

(d) At that frequency, what is the kinetic energy of an emerging proton?

Short answer

1. The magnitude of the magnetic field to achieve resonancewill be 0.787 T.
2. The kinetic energy of the protonwill be 8.34 x 10⁶ eV
3. 2.39 x 10⁷ Hz is the oscillator frequency to achieve resonance.
4. The kinetic energy of the emerging proton will be 3.32 x 10⁷eV

Step-by-step solution

Listing the given quantities

• $r=53.0cm=0.53m$
  
• $f=12.0MHz=12.0\times {10}^{6}Hz$
  
• $B=1.57T$
  

Understanding the relationship between the frequency and the magnetic field

We can find the relation between the frequency and the magnetic field using the relation$\mathit{q}\mathit{v}\mathit{B}\mathbf{=}\frac{\mathbf{m}{\mathbf{v}}^{\mathbf{2}}}{\mathbf{r}}$From this; we can solve for the B. We can find the equation for vin terms of f. We substitute the value for vin the formula of kinetic energy to get the kinetic energy of the proton.

We use the same relation, i.e.,$\mathit{q}\mathit{v}\mathit{B}\mathbf{=}\frac{\mathbf{m}{\mathbf{v}}^{\mathbf{2}}}{\mathbf{r}}$This time, we solve for f. We can find the equation for vin terms of f. We substitute the value for vin the formula of kinetic energy to get the kinetic energy of the proton emerging from the cyclotron.

Formula:

$\begin{array}{rcl}qvB& =& \frac{m{v}^2}{r}\\ K& =& \frac{1}{2}m{v}^2\\ \omega & =& v/r\\ \omega & =& 2\pi f\end{array}$

(a) Calculations of the magnitude of the magnetic field to achieve resonance

We know that:

$\begin{array}{rcl}qvB& =& \frac{m{v}^2}{r}\\ qB& =& \frac{mv}{r}\\ \frac{qB}{m}& =& \frac{v}{r}\\ \omega & =& \frac{qB}{m}\end{array}$

We know that:

$\begin{array}{rcl}\omega & =& 2\pi f\\ 2\pi f& =& \frac{qB}{m}\end{array}$

The magnitude of the magnetic field can be calculated as:

$\begin{array}{rcl}B& =& \frac{2\mathrm{\pi fm}}{q}\\ & =& \frac{2\mathrm{\pi }\times 12.0\times {10}^6\times 1.67\times {10}^{-27}}{1.6\times {10}^{-19}}\\ & =& 0.787T\end{array}$

The magnitude of the magnetic field to achieve resonance will be 0.787 T.

(b) Calculations of the kinetic energy of the proton

$K=\frac{1}{2}m{v}^2$

We know that

role="math" localid="1662880233360" $\begin{array}{rcl}\frac{v}{r}& =& omega\\ \frac{v}{r}& =& 2\mathrm{\pi f}\\ \mathrm{v}& =& 2\mathrm{\pi rf}\\ \mathrm{K}& =& \frac{1}{2}\times \mathrm{m}\times {\left(2\mathrm{\pi rf}\right)}^2\\ & =& \frac{1}{2}\times 1.67\times {10}^{-27}\times {\left(2\mathrm{\pi }\times 0.53\times 12\times {10}^6\right)}^2\\ & =& 1.33\times {10}^{-12}\mathrm{J}\\ & =& 1.33\times {10}^{-12}\mathrm{J}\times \frac{\left(6.24\times {10}^{18}\right)\mathrm{eV}}{1\mathrm{J}}\\ & =& 8.34\times {10}^6\mathrm{eV}\end{array}$

The kinetic energy of the proton will be$\begin{array}{rcl}& & 8.34\times {10}^6\mathrm{eV}\end{array}$

(c) Calculations of the oscillator frequency to achieve resonance

The frequency can be calculated as:

$\begin{array}{rcl}f& =& \frac{qB}{2\mathrm{\pi m}}\\ & =& \frac{1.6\times {10}^{-19}\times 1.57}{2\mathrm{\pi }\times 1.67\times {10}^{-27}}\\ & & \\ & =& 2.39\times {10}^7\mathrm{Hz}\end{array}$

$\begin{array}{rcl}& & 2.39\times {10}^7\mathrm{Hz}\end{array}$is the oscillator frequency to achieve resonance.

(d) Calculations of the kinetic energy of the emerging proton

The expression for kinetic energy:

$K=\frac{1}{2}m{v}^2$

We know that

$$\begin{gathered} \frac{v}{r}=omega \\ \frac{v}{r}=2\mathrm{\pi f} \end{gathered}$$

The expression for velocity can be written as:

$\begin{array}{rcl}v& =& 2\mathrm{\pi rf}\\ \mathrm{K}& =& \frac{1}{2}\times \mathrm{m}\times {\left(2\mathrm{\pi rf}\right)}^2\\ & =& \frac{1}{2}\times 1.67\times {10}^{-27}\times {\left(2\mathrm{\pi }\times 0.53\times 2.39\times {10}^7\right)}^2\\ & =& 5.3\times {10}^{-12}\mathrm{J}\\ & =& 5.3\times {10}^{-12}\mathrm{J}\times \frac{\left(6.24\times {10}^{-18}\right)\mathrm{eV}}{1\mathrm{J}}\\ & =& 3.32\times {10}^7\mathrm{eV}\end{array}$

The kinetic energy of the emerging proton will be$\begin{array}{rcl}& & 3.32\times {10}^7\mathrm{eV}\end{array}$