Question

How much energy is imparted to an electron so that its de-Broglie wavelength reduces from \( 10^{-10} \) m to \( 0.5 \times 10^{-10} \) m?

• A. \( 4E \)
• B. \( 2E \)
• C. \( 3E \)
• D. \( E \)

Hint:

The energy of an electron is related to its momentum. A change in momentum leads to a change in energy, and for de-Broglie wavelengths, this relationship is key.

Answer 1

The Correct Option is C

Step 1: Using the de-Broglie wavelength formula.
The de-Broglie wavelength \( \lambda \) is related to the momentum \( p \) of the electron by the equation: \[ \lambda = \frac{h}{p} \] Where \( h \) is Planck's constant. The energy of the electron is related to its momentum by: \[ E = \frac{p^2}{2m} \] Step 2: Using the change in wavelength.
Since the wavelength reduces by half, the momentum must increase by a factor of \( \sqrt{2} \). Therefore, the energy increases by a factor of 3. Thus, the correct answer is (C) \( 3E \).