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

The energy of a single photon is determined by the formula $E = \frac{hc}{\lambda}$, using the given wavelength of 6670 Angstroms and the standard constant value of 12400 eV-Angstroms for the numerator. Dividing the total emitted power of 9 mW by this individual photon energy determines the rate of photon arrival. Executing this division along with the necessary unit conversion factor yields an average of $3 \times 10^{16}$ photons arriving per second.

The stopping potential ($V_0$), which determines the maximum kinetic energy of emitted photoelectrons via the relation $e V_0 = K_{max}$, depends exclusively on the frequency of the incident radiation. Because curves (a) and (b) share the exact same stopping potential on the negative axis, the underlying radiations must have identical frequencies. Meanwhile, the different levels of saturation photocurrent shown by these two curves indicate that they possess varying intensities.

The emission of photoelectrons requires the incident photon energy to exceed the work function, a condition determined strictly by the threshold frequency. However, once this condition is met, the rate at which electrons are knocked out of the metal depends entirely on the rate of incoming photons. Since light intensity measures the density of these incident photons, the total number of emitted photoelectrons varies directly with the intensity of the light.

De Broglie wavelength is given by $\lambda = \frac{h}{mv}$, which indicates that equal wavelengths correspond to equal momenta for any two particles. Equating the momentum of the particle to that of the electron gives the relation $m_p v_p = m_e v_e$. Using the given mass for the particle along with the electron mass and velocity yields a particle velocity of 2.7 × 10^-18 ms^-1.

The total energy of the incident photon is obtained by adding the work function of 6.2 eV and the maximum kinetic energy corresponding to the stopping potential of 5 eV, yielding 11.2 eV. The corresponding wavelength is then calculated using the relation $\lambda = \frac{hc}{E}$, substituting 12400 eV-Angstroms for the numerator to find a wavelength of 1107 Angstroms. Because electromagnetic radiation with a wavelength around 1107 Angstroms lies below 4000 Angstroms and above X-rays, the radiation resides in the ultraviolet region.

Luminescent glows in gas discharge tubes arise from inelastic collisions between energetic charged particles emitted from the cathode and neutral atoms of the gas. These impacts excite or ionize the gas atoms, and the subsequent radiative de-excitation releases photons of characteristic visible colors.

Intensity of a spherical light source follows an inverse-square dependence on distance given by $I = \frac{P}{4 \pi d^2}$, meaning the ratio of intensities at two different positions is expressed as $\frac{I_1}{I_2} = \left(\frac{d_2}{d_1}\right)^2$. Since the rate of photoelectron emission is directly proportional to the incident light intensity, altering the position of the source reduces the emission rate by this same intensity factor. Evaluating this ratio for distances of 1.0 m and 0.5 m yields a reduction factor of 4.

The total power emitted by a monochromatic light source is related to the photon emission rate through the formula $P = N h \nu$, where $N$ represents the number of photons emitted per second and $h \nu$ is the energy of an individual photon. Rearranging this expression to solve for the emission rate gives $N = \frac{P}{h \nu}$. Substituting the given power, frequency, and Planck's constant yields an average emission rate of 5 × 10^15 photons per second.

A charged particle moving with a velocity perpendicular to a uniform magnetic field experiences a magnetic force that acts constantly orthogonal to its direction of motion, serving as the centripetal force. When the electric field is removed, this unbalanced magnetic force causes the electron to continuously curve into a uniform circular orbit.

The momentum of a photon is related to its energy through the formula $p = \frac{E}{c}$. Substituting the energy of 1 MeV converted into joules and dividing by the speed of light yields a value of 5.3 times 10 to the power -21 kg m/s, which rounds to approximately 5 times 10 to the power -22 kg m/s.

Gas ionization inside a discharge tube occurs via impact ionization, where free electrons accelerated by a high electric field gain sufficient kinetic energy to knock bound electrons out of neutral gas atoms or molecules. As these negative electrons collide with neutral particles, they liberate secondary electrons and create positive ions, sustaining the electrical discharge.

The magnitude of the photoelectric current is directly proportional to the intensity of the incident radiation, meaning more photons eject a greater number of electrons per second. In a photocell, this principle is utilized so that variations in light intensity directly translate into measurable fluctuations in the resulting electric current.

Einstein's photoelectric equation expresses the maximum kinetic energy of an emitted photoelectron as $K = h\nu - E_0$. When the frequency of the incident radiation is doubled to $2\nu$, the new maximum kinetic energy becomes $K' = h(2\nu) - E_0$. Expanding this expression yields $K' = 2h\nu - E_0$, which can be rearranged algebraically as $(h\nu - E_0) + h\nu$ to demonstrate that $K' = K + h\nu$.

Einstein's photoelectric equation is expressed as $\frac{1}{2} m v_{\max}^{2} = h \nu - W$. For the first condition where the photon energy is $2 h \nu_{0}$, the maximum kinetic energy corresponds to $h \nu_{0}$. When the photon energy is increased to $5 h \nu_{0}$, the new maximum kinetic energy becomes $4 h \nu_{0}$, which is four times the initial value. Since velocity scales with the square root of kinetic energy, multiplying the kinetic energy by four doubles the velocity, resulting in a maximum velocity of $8 \times 10^{6}$ m/s.

Photoelectric emission occurs only when the energy of the incident photon satisfies the condition $E \ge \phi$ relative to the work function of the metal. The energy of a photon corresponding to a wavelength of 4100 Å is approximately 3.02 eV when calculated, which exceeds the work functions of 1.92 eV and 2.0 eV while remaining below 5 eV.

Einstein's photoelectric equation relates the maximum kinetic energy of ejected photoelectrons to the frequency of incident radiation through the expression $K = h\nu - h\nu_0$, where $\nu_0$ represents the threshold frequency. Photoemission initiates only when the incident frequency surpasses this threshold, keeping the kinetic energy at zero beforehand. Once this limit is crossed, increasing the frequency produces a proportional rise in kinetic energy, yielding a straight line graph with a constant positive slope.

Cathode rays consist of subatomic particles with an exceptionally small mass relative to their charge, which J.J. Thomson determined by deflecting them using electric and magnetic fields. Because the mass of an electron is roughly two thousand times smaller than that of a proton, its charge-to-mass ratio is vastly greater. This precise measurement established that these corpuscles were fundamental components common to all matter rather than atoms or ions of the residual gas.

The intensity of light from a point source follows an inverse square law, meaning it is inversely proportional to the square of the distance from the source, expressed as $I \propto \frac{1}{d^{2}}$. When the distance is doubled from $1$ m to $2$ m, the light intensity drops to one-fourth of its initial value. Since the photoelectric current and the rate of photoelectron emission are directly proportional to the incident light intensity, the number of emitted electrons also reduces to a quarter of the initial number.

The de Broglie wavelength of a moving particle is inversely proportional to its momentum, given by the relation $\lambda = \frac{h}{mv}$. Since all the given particles travel with the same velocity, the particle possessing the smallest mass will yield the largest wavelength. Among a proton, neutron, alpha particle, and beta particle, the beta particle has by far the smallest mass, making its de Broglie wavelength the maximum.

Cathode rays consist of moving electrons, which are charged particles that experience a deflection when passing through an electric or magnetic field due to the electrostatic force exerted on them. Since they carry a negative charge, an external electric field readily alters their straight-line trajectory. Furthermore, these rays travel in straight lines to cast shadows, possess kinetic energy that generates heat upon striking matter, and cause certain materials to fluoresce. Claiming they do not deflect in an electric field directly contradicts their fundamental nature as charged particle beams.