An electron against a proton.

An electron and a proton have the same kinetic energy. The ratio of their de Broglie wavelengths (electron : proton) is about (the proton's mass is about 1836 times the electron's):

Answer: 43 : 1.

For a given kinetic energy K, the momentum is p = √(2mK), so the de Broglie wavelength λ = h/√(2mK) is inversely proportional to the square root of the mass.

\frac{λ_{e}}{λ_{p}} = \sqrt{\frac{m_{p}}{m_{e}}} = \sqrt{1836} \approx 42.8

The electron's wavelength is about 43 times longer. Using the mass ratio without the square root gives 1836, which would be right only if the two particles had the same speed rather than the same energy.

Original practice question · JEE Physics Decoded

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