Question

According to de-Broglie hypothesis, the ratio of the wave-length of a photon and that of an electron having same energy 'E' is (m = mass of electron, c = velocity of light)

• A. \(\frac{1}{C}\sqrt{\frac{E}{2m}}\)
• B. \(C\sqrt{\frac{E}{2m}}\)
• C. \(C\sqrt{\frac{2m}{E}}\)
• D. \(\sqrt{\frac{2mC}{E}}\)

Hint:

Photon: lambda = hc/E. Electron: lambda = h / sqrt(2 m E).

Answer 1

The Correct Option is C

Step 1: Check by dimensions:
$c\sqrt{2m/E}$: $\sqrt{m/E}$ has unit $\dfrac{\text{s}}{\text{m}}$, and multiplying by $c$ gives a pure number. Good, as a ratio of lengths must be.

Step 2: Check the others:
A and B contain $\sqrt{E/m}$, a speed, and A divides by $c$, giving a ratio of speeds, not of lengths. D has $\sqrt{mc/E}$ which is dimensionally wrong.

Step 3: Pick:
Option C.

Final Answer:
Dividing hc/E by h/sqrt(2mE) gives c sqrt(2m/E). \[ \boxed{\text{(C) }c\sqrt{\dfrac{2m}{E}}} \]