Magnetism
Intermediate
Electric Particles in Magnetic Field
Source: High school physics (Chinese)
Problem Sets:
magnetic field
Problem
A cyclotron has a uniform magnetic field of magnitude $B = 1.5$ T, and the dees have a maximum radius $R = 0.5$ m. Protons are accelerated in this cyclotron. The proton mass is $m = 1.67 \times 10^{-27}$ kg and its charge is $q = 1.6 \times 10^{-19}$ C.
- Find the cyclotron frequency.
- Find the maximum kinetic energy of the protons as they leave the accelerator.
$f \approx 2.29 \times 10^{7}$ Hz ($\approx 22.9$ MHz); $E_{\max} \approx 4.31 \times 10^{-12}$ J $\approx 27$ MeV.
The cyclotron period is $T = \dfrac{2\pi m}{qB}$ and the cyclotron frequency is independent of speed:
$$f = \dfrac{1}{T} = \dfrac{qB}{2\pi m}.$$(1) $f = \dfrac{(1.6 \times 10^{-19})(1.5)}{2\pi (1.67 \times 10^{-27})} \approx 2.29 \times 10^{7}$ Hz $\approx 22.9$ MHz.
(2) At maximum radius, $v_{\max} = \dfrac{qBR}{m}$, so
$$E_{\max} = \dfrac{1}{2} m v_{\max}^2 = \dfrac{q^2 B^2 R^2}{2m} = \dfrac{(1.6 \times 10^{-19})^2 (1.5)^2 (0.5)^2}{2 (1.67 \times 10^{-27})} \approx 4.31 \times 10^{-12} \text{ J} \approx 27 \text{ MeV}.$$