A photon has energy \( E \), and a proton moving at a speed much smaller than that of light has kinetic energy \( E \). Match the quantities in List-I with their dependence on \( E \) in List-II.
| List-I | List-II |
| (A) Wavelength of the photon | (I) Proportional to \( E \) |
| (B) Momentum of the photon | (II) Proportional to \( E^{1/2} \) |
| (C) de Broglie wavelength of the proton | (III) Proportional to \( E^{-1/2} \) |
| (D) Momentum of the proton | (IV) Proportional to \( E^{-1} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(I), (C)-(III), (D)-(II)
A photon of frequency \( \nu \) has energy \( E=h\nu \) and momentum \( p=h\nu/c \). For the slow proton, the kinetic energy is \( E=p^{2}/2m \). Its de Broglie wavelength is \( \lambda=h/p \).
(A) Wavelength of the photon — This matches (IV) Proportional to \( E^{-1} \). From \( E=hc/\lambda \), the wavelength is \( \lambda=hc/E \), so it falls as the energy rises.
(B) Momentum of the photon — This matches (I) Proportional to \( E \). The momentum is \( p=h\nu/c \), and since \( h\nu \) is the energy, \( p=E/c \).
(C) de Broglie wavelength of the proton — This matches (III) Proportional to \( E^{-1/2} \). Putting \( p=\sqrt{2mE} \) in \( \lambda=h/p \),
\( \displaystyle \lambda=\frac{h}{\sqrt{2mE}} \)
(D) Momentum of the proton — This matches (II) Proportional to \( E^{1/2} \). From \( E=p^{2}/2m \), the momentum is \( p=\sqrt{2mE} \).
NCERT (2026-27):
Particle Nature of Light: The Photon – “(iii) All photons of light of a particular frequency ν, or wavelength λ, have the same energy E (=hν = hc/λ) and momentum p (= hν/c= h/λ), whatever the intensity of radiation may be.”
Wave Nature of Matter – “… De Broglie proposed that the wave length λ associated with a particle of momentum p is given as
\( \displaystyle \lambda=\frac{h}{p} \)\( \displaystyle =\frac{h}{mv} \qquad (11.5) \)
where m is the mass of the particle and v its speed.”