Question

The wavelengths of a proton and a photon are the same. Then:
A. Their velocities are equal
B. Their momenta are equal
C. Their energy are equal
D. Their speed are equal

Answer 1

The wavelength of a periodic wave is its spatial period, or the distance over which the wave's form repeats. It's the distance between two adjacent corresponding points of the same phase on the wave, such two adjacent crests, troughs, or zero crossings, and it's a feature of both travelling and standing waves, as well as other spatial wave patterns.

Complete step by step answer:
A wave's wavelength is determined by the medium through which it travels (for example, vacuum, air, or water). Sound waves, light waves, water waves, and periodic electrical impulses in a conductor are all examples of waves. A sound wave is a change in air pressure, whereas the intensity of the electric and magnetic fields varies in light and other electromagnetic radiation. Variations in the height of a body of water are known as water waves. Atomic locations change in a crystal lattice vibration.

The connection between mass and energy in a system's rest frame, when the two quantities differ only by a constant and the units of measurement, is known as mass–energy equivalence in physics. The famous formula of physicist Albert Einstein describes the principle:
$E = m{c^2}$
Now for proton,
$\lambda = \dfrac{{\text{h}}}{{{\text{mv}}}} \\\$
$\Rightarrow {\lambda _1} = \dfrac{{\mathbf{h}}}{{\mathbf{p}}}$
For photon,
${\mathbf{E}} = \dfrac{{{\mathbf{hc}}}}{\lambda }$
\Rightarrow {\lambda _2} = \dfrac{{\text{h}}}{{{\text{E}}{{\text{c}}^{ - 1}}}} \\\
$\Rightarrow {\lambda _1} = {\lambda _2}$............(Given)
$\Rightarrow \dfrac{{\mathbf{h}}}{{\mathbf{p}}}=\dfrac{{\text{h}}}{{{\text{E}}{{\text{c}}^{ - 1}}}} \\\$
$\therefore {\text{p}} = \dfrac{{\text{E}}}{{\text{c}}}$
Hence momentum is constant.

Thus, option B is correct.

Note: Wavelength depends on the medium of light but frequency is independent of medium of light because it is the fundamental property of light. Two photons of the same frequency will have the same energy. The higher the frequency of light , the higher will be the energy and hence higher will be their amplitude.