Question

The energy that should be added to an electron to reduce its de-broglie wavelength from 1 nm to 0.5 nm is

A) Four times the initial energy
B) Equal to the initial energy
C) Twice the initial energy
D) Thrice the initial energy

Answer 1

The energy needed to reduce the de Broglie wavelength of an electron from 1 nm to 0.5 nm is four times the initial energy because kinetic energy is proportional to the square of momentum.

Step-by-step explanation:

The question pertains to the concept of the de Broglie wavelength of an electron and the relationship between its wavelength, momentum, and kinetic energy. According to the de Broglie hypothesis, the wavelength (λ) of a particle is inversely proportional to its momentum (p), which can be expressed as λ = h/p, where h is Planck's constant.

When an electron's de Broglie wavelength decreases, this implies an increase in its momentum and hence its kinetic energy. Given that kinetic energy (K) is related to momentum by the equation K = p²/2m (where m is the mass of the electron), if the wavelength is halved, the momentum is doubled. Since the kinetic energy is proportional to the square of the momentum, if the momentum doubles, the kinetic energy increases by a factor of four.

Therefore, the energy that should be added to an electron to reduce its de Broglie wavelength from 1 nm to 0.5 nm is four times the initial energy, which makes option (A) the correct answer.