Question

Choose correct option :
(A) Number of photons required for a light beam of 2000 pm wavelength and 1 Joule energy is \(1.01 \times 10^{16}\).
(B) Light with wavelength 300 nm has energy \(E_1\) and for wavelength 900 nm has energy \(E_2\), then \(E_1/E_2 = 1/3\).
(C) Frequency of light is \(4.5 \times 10^{16}\) Hz then its wavelength is \(6.7 \times 10^{-9}\) m.
(D) If electrons and protons are accelerated by same potential difference, then their de-Broglie wavelength are equal.

• A. A only
• B. A & B only
• C. A & C only
• D. A, B, C & D

Hint:

For "multiple correct" or "choose the correct statement" questions, evaluate each option systematically and independently.
Remember the relationship \(\lambda = h/p\) and how momentum \(p\) relates to kinetic energy \(K\) (\(p=\sqrt{2mK}\)) and accelerating potential \(V\) (\(K=qV\)). This is a very common topic.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to evaluate the correctness of four separate statements related to photons, electromagnetic waves, and de-Broglie wavelength. 
Step 2: Detailed Explanation: 
Statement (A): 
Energy of a single photon is \(E_{ph} = \frac{hc}{\lambda}\). 
The number of photons \(N\) in a beam of total energy \(E_{total}\) is \(N = \frac{E_{total}}{E_{ph}} = \frac{E_{total} \lambda}{hc}\). 
Given: \(E_{total} = 1\) J, \(\lambda = 2000 \text{ pm} = 2000 \times 10^{-12}\) m. 
\(h = 6.626 \times 10^{-34}\) J·s, \(c = 3 \times 10^8\) m/s. 
\[ N = \frac{(1 \text{ J})(2000 \times 10^{-12} \text{ m})}{(6.626 \times 10^{-34} \text{ Js})(3 \times 10^8 \text{ m/s})} = \frac{2 \times 10^{-9}}{19.878 \times 10^{-26}} \approx 0.1006 \times 10^{17} = 1.006 \times 10^{16} \] This value is very close to \(1.01 \times 10^{16}\). Statement (A) is correct. 
Statement (B): 
The energy of a photon is inversely proportional to its wavelength (\(E \propto 1/\lambda\)). 
Therefore, the ratio of energies is \(\frac{E_1}{E_2} = \frac{\lambda_2}{\lambda_1}\). 
Given: \(\lambda_1 = 300\) nm, \(\lambda_2 = 900\) nm. 
\[ \frac{E_1}{E_2} = \frac{300 \text{ nm}}{900 \text{ nm}} = 3 \] The statement claims \(\frac{E_1}{E_2} = 1/3\). Statement (B) is incorrect. 
Statement (C): 
The relationship between frequency (\(f\)), wavelength (\(\lambda\)), and the speed of light (\(c\)) is \(\lambda = \frac{c}{f}\). 
Given: \(f = 4.5 \times 10^{16}\) Hz. 
\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{4.5 \times 10^{16} \text{ Hz}} = \frac{3}{4.5} \times 10^{-8} \text{ m} = \frac{2}{3} \times 10^{-8} \text{ m} \approx 0.667 \times 10^{-8} \text{ m} = 6.67 \times 10^{-9} \text{ m} \] This value is approximately \(6.7 \times 10^{-9}\) m. Statement (C) is correct. 
Statement (D): 
The de-Broglie wavelength is given by \(\lambda = \frac{h}{p}\), where \(p\) is the momentum. 
When a particle of charge \(q\) is accelerated by a potential difference \(V\), its kinetic energy is \(K = qV\). 
Momentum is related to kinetic energy by \(p = \sqrt{2mK}\). 
So, \(\lambda = \frac{h}{\sqrt{2mK}} = \frac{h}{\sqrt{2mqV}}\). 
Electrons and protons have the same magnitude of charge \(q\) but different masses (\(m_p >> m_e\)). If they are accelerated by the same \(V\), their wavelengths will be different. Specifically, since mass is in the denominator, the more massive proton will have a shorter wavelength. Statement (D) is incorrect. 
Step 3: Final Answer: 
Statements (A) and (C) are correct, while (B) and (D) is incorrect. The best choice among the options is (C) which states A & C are correct.