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

Let's evaluate each option to determine the correct answer:

1. Option (A): Number of photons required for a light beam of 2000 pm wavelength and 1 Joule energy is \(1.01 \times 10^{16}\).
   • The energy of a single photon is given by the formula: \(E = \frac{hc}{\lambda}\), where \(h = 6.626 \times 10^{-34}\) Js\) (Planck's constant), \(c = 3 \times 10^{8} \, m/s\) (speed of light), and \(\lambda = 2000 \times 10^{-12} \, m\) (since 1 pm = \(10^{-12}\) m).
   • Calculate the photon energy: \(E = \frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{2000 \times 10^{-12}}\) 
   • Solve to find \(E \approx 9.939 \times 10^{-17} \, J\).
   • Number of photons \(n = \frac{1 \, J}{9.939 \times 10^{-17} \, J/photon}\) which gives approximately \(1.01 \times 10^{16}\) photons.
   • This option is correct.
2. Option (B): Light with wavelength 300 nm has energy \(E_1\) and for wavelength 900 nm has energy \(E_2\), then \(\frac{E_1}{E_2} = \frac{1}{3}\).
   • The energy of a photon is inversely proportional to its wavelength: \(E \propto \frac{1}{\lambda}\).
   • For 300 nm: \(E_1 \propto \frac{1}{300 \, nm}\) and for 900 nm: \(E_2 \propto \frac{1}{900 \, nm}\).
   • The ratio \(\frac{E_1}{E_2} = \frac{\frac{1}{300}}{\frac{1}{900}} = 3\), not \(\frac{1}{3}\).
   • This option is incorrect.
3. Option (C): Frequency of light is \(4.5 \times 10^{16} \, Hz\) then its wavelength is \(6.7 \times 10^{-9} \, m\).
   • The relationship between frequency and wavelength is given by \(c = \lambda \nu\), where \(\nu\) is the frequency.
   • Calculate wavelength \(\lambda = \frac{c}{\nu} = \frac{3 \times 10^{8}}{4.5 \times 10^{16}}\)
   • Solve to find \(\lambda = 6.7 \times 10^{-9} \, m\).
   • This option is correct.
4. Option (D): If electrons and protons are accelerated by the same potential difference, then their de-Broglie wavelengths are equal.
   • The de-Broglie wavelength is given by \(\lambda = \frac{h}{\sqrt{2mVq}}\), where \(m\) is mass and \(q\) is charge.
   • Under the same potential difference, the electrons and protons have different masses, leading to different wavelengths.
   • Therefore, this statement is incorrect.

Thus, the correct answer is the options A & C only.