What is the de Broglie wavelength of an electron that has been accelerated through a potential difference of 100 V?

###### Solution.

The electron accelerated through the potential difference gains kinetic energy.

Let’s write the equation of conservation of energy

eU = 0.5mv^{2} ,

where U – the potential difference, e – the charge of the electron 1.6*10^{-19} C, m – the mass of electron 9.11*10^{-31} kg.

Thus, the velocity will be

v = √(2eU/m) = √(2*1.6*10^{-19}*100/9.11*10^{-31}) ≈ 5.93*10^{6} m/s.

De Broglie wavelength of an electron is given by the equation

λ = h/(mv) ,

where h is Planck’s constant 6.62*10^{-34} J*s.

Thus,

λ = 6.62*10^{-34} /(9.11*10^{-31}*5.93*10^{6}) ≈ 1.2*10^{-10} m = 1.2 Å

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What will be the amount of energy radiated in unit time if that electron is accelerated? I am talking in accordance with classical theory of electromagnetism.

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