MHT CET · Physics · Dual Nature of Matter

What is the additional energy that should be supplied to a moving electron to reduce its de Broglie wavelength from \(1 \mathrm{~nm}\) to \(0.5 \mathrm{~nm}\) ?

A Four times its initial energy
B Five times its initial energy
C Two times its initial energy
D Three times its initial energy

Verified Solution

Answer & Solution

Correct Answer

(D) Three times its initial energy

Step-by-step Solution

Detailed explanation

De Broglie wavelength, \(\lambda=\frac{\mathrm{h}}{\mathrm{mv}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mE}}}\)
\(
\Rightarrow \mathrm{E}=\frac{\mathrm{h}^2}{2 \mathrm{~m} \lambda^2}
\)
If \(\lambda\) is halved, then kinetic energy becomes 4 times or increased by three times.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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