Question

The wavelength $\lambda_{e}$ of an electron and $\lambda_{p}$ of a photon of same energy $\mathbf{E}$ are related by
(A) $\lambda_{p} \propto \lambda_{c}^{2}$
(B) $\lambda_{p} \propto \lambda_{e}$
(C) $\lambda_{p} \propto \sqrt{\lambda_{c}}$
(D) $\lambda_{p} \propto \dfrac{1}{\sqrt{\lambda_{c}}}$

Answer 1

Hint: We know that protons and electrons stick to each other as much as they can, but kinetic energy and quantum mechanics keep them from holding still. Protons and electrons are attracted to each other because the positive electric charge of the proton is attracted to the negative charge of the electron. Unlike protons and neutrons, which are located inside the nucleus at the center of the atom, electrons are found outside the nucleus. Because opposite electric charges attract each other, negative electrons are attracted to the positive nucleus.

Complete step by step answer
We know that an electron is a negatively charged subatomic particle electron that can be either free (not attached to any atom), or bound to the nucleus of an atom. In conductors current flows due to the movement of electrons. Electrons are the negatively charged particles of atoms. Together, all of the electrons of an atom create a negative charge that balances the positive charge of the protons in the atomic nucleus. Electrons are extremely small compared to all of the other parts of the atom.
We know that:
Wavelength of electron, $\lambda_{\mathrm{e}}=\dfrac{\mathrm{h}}{\sqrt{2 \mathrm{mE}}}$
Now we can say that:
Wavelength of photon, $\lambda_{\mathrm{p}}=\dfrac{\mathrm{hc}}{\mathrm{E}}$
$\lambda_{\mathrm{e}}^{2}=\dfrac{\mathrm{h}^{2}}{2 \mathrm{mE}}$
$\lambda_{\mathrm{e}}^{2}=\dfrac{\mathrm{h}^{2}}{2 \mathrm{m} \dfrac{\mathrm{hc}}{\lambda_{\mathrm{p}}}}$
$\Rightarrow \lambda_{\mathrm{e}}^{2} \propto \lambda_{\mathrm{p}}$

We know that the correct option is option A.

Note: Electrons are not little balls that can fall into the nucleus under electrostatic attraction. Rather, electrons are quantized wave functions that spread out in space and can sometimes act like particles in limited ways. An electron in an atom spreads out according to its energy. Electron hits Proton and disintegrates it. Electrons hit Proton and bounce off each other. Electrons get sucked in Proton and form a neutral particle Neutron. It is called Inverse Beta Decay or Electron Capture. An electron is called a universal particle because the value of charge and mass of the electron remains constant whatever may be the material. Electron is called a universal particle because charge and mass of the electron remain the same in every material.