Dual Nature Of Matter And Radiation – NEET UG PhysicsPractice Questions & PYQs

Planck's constant is a fundamental physical constant that relates the energy of a quantum of electromagnetic radiation to its frequency. Its experimentally determined magnitude is universally recognized in quantum mechanics as $6.63 \times 10^{-34}$ joule-seconds. This dimensional product of energy and time forms the basic scaling factor for quantum actions across atomic and subatomic systems.

The photoelectric effect depends on the frequency of incident radiation exceeding a specific threshold frequency, which corresponds to having a wavelength shorter than the threshold wavelength. If ultraviolet light fails to eject electrons from a given metal, the metal's threshold wavelength is shorter than that of ultraviolet rays, necessitating radiation of even higher frequency and shorter wavelength. Among the available choices, X-rays possess a shorter wavelength and higher energy than ultraviolet rays, allowing them to successfully overcome the work function and initiate electron emission. Infrared rays, radio waves, and microwaves all have longer wavelengths and lower energies than ultraviolet light.

A charged particle passes undeflected through perpendicular electric and magnetic fields when the electric force balances the magnetic force. Equating the electric force $F_e = e|\vec{E}|$ with the magnetic force $F_m = ev|\vec{B}|$ provides the condition for straight-line motion without deviation. Rearranging this balance equation yields the velocity expression $v = \frac{|\vec{E}|}{|\vec{B}|}$, which selects particles possessing this specific speed regardless of their mass or charge.

The particle nature of light is manifested in interactions where energy is exchanged in discrete packets or quanta rather than as a continuous wavefront. Phenomena like interference, refraction, and polarization rely on the superposition and bending of continuous wave fronts, successfully explained entirely by wave theory. In contrast, the photoelectric effect requires light to behave as a stream of individual photons colliding with electrons and transferring their energy instantaneously. This corpuscular interaction cannot be accounted for by wave models, thereby proving the particulate behavior of light.

The intensity of light from a point source drops off inversely with the square of the distance, following the relation $I \propto \frac{1}{d^2}$. When the distance is halved to $\frac{d}{2}$, the new intensity becomes $I' \propto \frac{1}{(d/2)^2}$, which simplifies to four times the original intensity. Since the number of photoelectrons emitted per second varies directly with the intensity of the illumination, the emission rate increases by a factor of four.

The specific charge, or charge-to-mass ratio, of cathode rays was first measured by J.J. Thomson through deflection experiments in electric and magnetic fields. By establishing this foundational ratio, the actual mass of the electron could be determined indirectly once the magnitude of the electronic charge was later discovered. Consequently, Thomson is credited with this crucial indirect evaluation that shaped our understanding of subatomic particles.

The de Broglie wavelength of a particle is inversely proportional to its momentum, given by $\lambda = \frac{h}{p}$, where momentum under an accelerating potential is expressed as $p = \sqrt{2meV}$. Setting up the ratio between the proton's wavelength and the electron's wavelength substitutes their respective masses while keeping the charge and potential difference constant. Simplifying this ratio yields the final expression $\lambda_p = \lambda \sqrt{\frac{m}{M}}$ for the proton.

The work done by an electric field on a charged particle appears entirely as its kinetic energy, which is expressed by the energy relation $eV = \frac{1}{2}mv^2$. Rearranging this fundamental formula to solve for the final speed $v$ isolates the velocity term, resulting in $v = \sqrt{\frac{2eV}{m}}$.

According to the theory of special relativity, an object's mass increases with its velocity relative to an observer, quantified by the Lorentz factor formula $m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$. Substituting the given speed $v = 0.8c$ into this expression gives $m = \frac{m_0}{\sqrt{1 - \frac{(0.8c)^2}{c^2}}}$. Evaluating the square root term results in $\sqrt{1 - 0.64}$, which simplifies to $\sqrt{0.36}$ or $0.6$. Dividing the rest mass by this fraction yields $\frac{5m_0}{3}$.

The de Broglie wavelength associated with a moving particle is calculated using the relation $\lambda = \frac{h}{\sqrt{2mE}}$. Converting the given kinetic energy into SI units by multiplying 100 eV by the conversion factor yields the total energy in joules. Substituting Planck's constant, the electron mass, and this energy into the formula produces a value of $1.2 \times 10^{-10}$ meters, which simplifies directly to $1.2$ angstroms.

The speed of light in a vacuum is a universal fundamental constant that remains entirely independent of frequency or wavelength. While the wave relation $c = \nu\lambda$ connects frequency and wavelength for any given wave, the velocity itself does not scale proportionally with frequency, rendering all standard proportionality choices incorrect.

The energy of a photon is directly proportional to its frequency, expressed through the fundamental relation $E = h \nu$, where $h$ is Planck's constant. Einstein utilized this principle to explain the photoelectric effect, demonstrating that light behaves as discrete packets of energy rather than a continuous wave. This quantum model successfully established that the emission of electrons depends entirely on the frequency of the incident radiation rather than its intensity.