Question

Accelerating charges radiate electromagnetic waves. calculate the wavelength of radiation produced by a proton in a cyclotron with a magnetic field of 0.400 t.

Answer 1

The wavelength of radiation produced by a proton in a cyclotron with a magnetic field of 0.400 t is $49.19$

Explanation:

Electromagnetic radiation is the electromagnetic field waves that propagates through space and carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, light, ultraviolet, X-rays, and gamma rays.

A proton is a subatomic particle with symbol p or p⁺ with positive electric charge of +1e elementary charge and a mass slightly less than neutron.  

A cyclotron  a type of particle accelerator that accelerates charged particles outwards from the center along a spiral path.

Accelerating charges radiate electromagnetic waves. calculate the wavelength of radiation produced by a proton in a cyclotron with a magnetic field of 0.400 t.

• B = magnetic field in the cyclotron = 0.400 T
• q = magnitude of charge on a proton = 1.6 x 10⁻¹⁹ C
• m = mass of the proton = 1.67 x 10⁻²⁷ kg
• f = frequency of revolution of proton in the cyclotron
• v = speed of electromagnetic waves = 3 x 10⁸ m/s

$\omega=\frac{v}{\rho} = \frac{qB}{m} = \frac{1.6*10^{-19}C * 0.4t}{1.67*10^{-27} kg}\\\omega = \frac{0.64*10^{-19} Ct }{1.67*10^{-27} kg} = 0.383 * 10^{8}$

$T = \frac{2*\pi}{\omega} = \frac{2*\pi}{0.383*10^{8}} = 16.396*10^{-8}$

$\omega = \frac{2\pi c}{\lambda} \\ 0.383*10^8= \frac{2\pi * 3*10^{8}}{\lambda} \\\lambda = \frac{18.84}{0.383} = 49.19$

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Answer 2

B = magnetic field in the cyclotron = 0.400 T

q = magnitude of charge on a proton = 1.6 x 10⁻¹⁹ C

m = mass of the proton = 1.67 x 10⁻²⁷ kg

f = frequency of revolution of proton in the cyclotron = ?

v = speed of electromagnetic waves = 3 x 10⁸ m/s

λ = wavelength of electromagnetic wave = ?

Frequency of revolution of proton in the cyclotron is given as

f = qB/(2πm)

inserting the values

f = (1.6 x 10⁻¹⁹)(0.400)/(2 (3.14) (1.67 x 10⁻²⁷))

f = 6.1 x 10⁶ Hz

wavelength of electromagnetic wave is given as

λ = v/f

λ = (3 x 10⁸)/(6.1 x 10⁶)

λ = 49.2 m