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Two electrons move with the same speed v. One enters a region of uniform electric field, the other enters a region of uniform magnetic field. After some time, if their de Broglie wavelengths are lambda1 and lambda2 respectively, which option(s) are possible?

1. (A) lambda1 = lambda2
2. (B) lambda1 > lambda2
3. (C) lambda1 < lambda2
4. (D) lambda1 > lambda2 or lambda1 < lambda2

Correct answer: (B) lambda1 > lambda2

Solution

For the electron in the magnetic field: the magnetic force is always perpendicular to the velocity, so it does no work. The speed (and hence momentum) of the electron remains v. Therefore lambda2 = h/(mv) = initial wavelength. For the electron in the electric field: the electric field does work on the electron, changing its kinetic energy and hence its speed. Depending on the field direction relative to initial velocity, the electron can speed up or slow down. If it speeds up, momentum increases, lambda1 < lambda2. If it slows down, momentum decreases, lambda1 > lambda2. Thus lambda1 can be greater than, equal to, or less than lambda2. Options A, B, C are all individually possible (D encompasses B and C but is not a single definitive answer). The question asks for 'possible options', so A, B, and C are all possible.

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