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An electron moving with speed v and a photon moving with speed c have the same de Broglie wavelength. Let the kinetic energy and momentum of the electron be Eₑ and Pₑ, and those of the photon be Eₚ and Pₚ. Which statement is correct?

1. Eₑ / Eₚ = 2c / v
2. Pₑ / Pₚ = 2c / v
3. Eₚ / Eₑ = 2c / v
4. Pₚ / Pₑ = 2c / v

Correct answer: Eₚ / Eₑ = 2c / v

Solution

Same wavelength => same momentum: Pₑ = Pₚ = p. Electron kinetic energy (non-relativistic): Eₑ = p²/(2mₑ). Since p = mₑ*v (for non-relativistic electron), Eₑ = mₑ*v²/2. Photon energy: Eₚ = p*c. Ratio Eₚ/Eₑ = p*c / (p²/(2mₑ)) = 2mₑ*c/p = 2mₑ*c/(mₑ*v) = 2c/v. Since Pₑ = Pₚ, any ratio of momenta = 1, so options B and D are wrong. The correct statement is Eₚ/Eₑ = 2c/v.

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