An electron and an alpha particle enter a uniform electric field with the same initial de Broglie wavelength ₀ . The initial velocities of both particles are perpendicular to the direction of the electric field. After a time t , the de Broglie wavelength of the electron becomes ₀ 2 . The de Broglie wavelength of the alpha particle at this same time t will be:

Step-by-step solution

The initial de Broglie wavelength is ₀ for both particles, which means they have the same initial momentum p₀ . The force on a particle of charge q in an electric field E is F = qE , acting perpendicular to the initial velocity. The change in momentum in the transverse direction after time t is p = qEt . For the electron (charge magnitude e ), the total momentum at time t is: p_e = p₀^2 + (eEt)^2 Given that _e = ₀ 2 , we have p_e = 2 p₀ . Squaring both sides: p₀^2 + (eEt)^2 = 2p₀^2 eEt = p₀ For the alpha particle (