Question

A cyclotron has a diameter of 25cm and accelerates protons at 1Mev

calculate speed of these protons
calculate momentum of these protons
calculate the magnetic field strength assuming that the protons move around the edge of the machine
calculate the frequency

Answer 1

Answer on Question #63736, Physics / Atomic and Nuclear Physics

A cyclotron has a diameter of 25cm and accelerates protons at 1Mev

a) calculate speed of these protons

b) calculate momentum of these protons

c) calculate the magnetic field strength assuming that the protons move around the edge of the machine

d) calculate the frequency

Solution:

$$E = 1 \mathrm{MeV} = 1 * 10^{6} \times 1.6 * 10^{-19} = 1.6 * 10^{-13} \mathrm{J}, \quad m = 1.67 * 10^{-27} \mathrm{kg}, \quad q = 1.6 \times 10^{-19} \mathrm{C}, \quad r = 2.5 * 10^{-1} \mathrm{m}$$

a) $E = m v^{2} / 2$

$$v = \sqrt{2} \times 1.6 * 10^{-13} \mathrm{J} / 1.67 * 10^{-27} \mathrm{kg} = 1.38 * 10^{7} \mathrm{m/s}$$

b) $p = m v$

$$p = 1.38 * 10^{7} \mathrm{m/s} \times 1.67 * 10^{-27} \mathrm{kg} = 2.31 * 10^{20} \mathrm{m/kg/s}$$

c) $B = m v / r q$

$$B = 1.67 * 10^{-27} \mathrm{kg} \times 1.38 * 10^{7} \mathrm{m/s} / 2.5 * 10^{-1} \mathrm{m} \times 1.6 \times 10^{-19} \mathrm{C} = 5.7 * 10^{-1} \mathrm{T}$$

d) $v = B q / 2 \pi m$

$$v = 5.7 * 10^{-1} \mathrm{T} \times 1.6 * 10^{-19} \mathrm{C} / 2 \times 3.14 \times 1.67 * 10^{-27} \mathrm{kg} = 8.82 * 10^{6} \mathrm{1/s}$$

Answer: a) $1.38 * 10^{7} \mathrm{m/s}$; b) $2.31 * 10^{20} \mathrm{m/kg/s}$; c) $5.7 * 10^{-1} \mathrm{T}$; d) $8.82 * 10^{6} \mathrm{1/s}$

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