Dual Nature of Radiation and Matter

The kinetic energy of emitted electron is E when the light incident on the metal has wavelength λ. To double the kinetic energy, the incident light must have wavelength :

An α particle and a proton are accelerated from rest through the same potential difference. The ratio of linear momenta acquired by above two particals will be:

The de Broglie wavelength associated with an electron, accelerated through a potential difference of 15056 V, is:

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals $P$, $Q$ and $R$ are $E_P$, $E_Q$ and $E_R$, respectively, and they are related by $E_P = 2E_Q = 2E_R$. In this experiment, the same source of monochromatic light is used for metals $P$ and $Q$ while a different source of monochromatic light is used for the metal $R$. The work functions for metals $P$, $Q$ and $R$ are 4.0 eV, 4.5 eV and 5.5 eV, respectively. The energy of the incident photon used for metal $R$, in eV, is ____.

Given below are two statements :

Statement I : Two photons having equal linear momenta have equal wavelengths.

Statement II : If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease.

In the light of the above statements, choose the correct answer from the options given below.

The filament of a light bulb has surface area $64\ \mathrm{mm^2}$. The filament can be considered as a black body at temperature 2500 K emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of 100 m. Assume the pupil of the eyes of the observer to be circular with radius 3 mm. Then

(Take Stefan-Boltzmann constant $= 5.67\times10^{-8}\ \mathrm{W\,m^{-2}K^{-4}}$, Wien's displacement constant $= 2.90\times10^{-3}$ m-K, Planck's constant $= 6.63\times10^{-34}$ Js, speed of light in vacuum $= 3.00\times10^{8}\ \mathrm{m\,s^{-1}}$)

A free hydrogen atom after absorbing a photon of wavelength $\lambda_a$ gets excited from the state $n = 1$ to the state $n = 4$. Immediately after that the electron jumps to $n = m$ state by emitting a photon of wavelength $\lambda_e$. Let the change in momentum of atom due to the absorption and the emission are $\Delta p_a$ and $\Delta p_e$, respectively. If $\lambda_a/\lambda_e = \frac15$, which of the option(s) is/are correct?

[Use $hc = 1242$ eV nm; 1 nm $= 10^{-9}$ m, $h$ and $c$ are Planck's constant and speed of light, respectively]

A perfectly reflecting mirror of mass $M$ mounted on a spring constitutes a spring-mass system of angular frequency $\Omega$ such that $\dfrac{4\pi M\Omega}h = 10^{24}\ \mathrm{m^{-2}}$ with $h$ as Planck's constant. $N$ photons of wavelength $\lambda = 8\pi\times10^{-6}$ m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 μm. If the value of $N$ is $x\times10^{12}$, then the value of $x$ is ______.

[Consider the spring as massless]

In a photoelectric experiment a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 eV. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force $F = n \times 10^{-4}$ N due to the impact of the electrons. The value of $n$ is __________. Mass of the electron $m_e = 9 \times 10^{-31}$ kg and $1.0\ \mathrm{eV} = 1.6 \times 10^{-19}$ J.

A photoelectric material having work-function $\phi_0$ is illuminated with light of wavelength $\lambda$ $\left(\lambda < \frac{hc}{\phi_0}\right)$. The fastest photoelectron has a de Broglie wavelength $\lambda_d$. A change in wavelength of the incident light by $\Delta\lambda$ results in a change $\Delta\lambda_d$ in $\lambda_d$. Then the ratio $\Delta\lambda_d/\Delta\lambda$ is proportional to

A particle A of mass m and initial velocity $v$ collides with a particle B of mass $\dfrac m2$ which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths $\lambda_A$ to $\lambda_B$ after the collision is :

The maximum velocity of the photoelectrons emitted from the surface is v when light of frequency n falls on a metal surface. If the incident frequency is increased to 3n, the maximum velocity of the ejected photoelectrons will be :

A Laser light of wavelength 660 nm is used to weld Retina detachment. If a Laser pulse of width 60 ms and power 0.5 kW is used the approximate number of photons in the pulse are :

[Take Planck's constant $h=6.62\times10^{-34}$ Js]

In a historical experiment to determine Planck's constant, a metal surface was irradiated with light of different wavelengths. The emitted photoelectron energies were measured by applying a stopping potential. The relevant data for the wavelength $(\lambda)$ of incident light and the corresponding stopping potential $(V_0)$ are given below:

$\lambda$ (μm)	$V_0$ (Volt)
0.3	2.0
0.4	1.0
0.5	0.4

Given that $c = 3\times10^{8}\ \mathrm{m\,s^{-1}}$ and $e = 1.6\times10^{-19}$ C, Planck's constant (in units of J s) found from such an experiment is

Light of wavelength $\lambda_{ph}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\phi$ and the anode is a wire mesh of conducting material kept at a distance $d$ from the cathode. A potential difference $V$ is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is $\lambda_e$, which of the following statement(s) is(are) true?

Radiation of wavelength λ, is incident on a photocell. The fastest emitted electron has speed $v$. If the wavelength is changed to $\dfrac{3\lambda}4$, the speed of the fastest emitted electron will be :

When photons of wavelength $\lambda_1$ are incident on an isolated sphere, the corresponding stopping potential is found to be V. When photons of wavelength $\lambda_2$ are used, the corresponding stopping potential was thrice that of the above value. If light of wavelength $\lambda_3$ is used then find the stopping potential for this case :

A photoelectric surface is illuminated successively by monochromatic light of wavelengths λ and $\dfrac\lambda2$. If the maximum kinetic energy of the emitted photoelectrons in the second case is 3 times that in the first case, the work function of the surface is :

Consider a hydrogen atom with its electron in the $n^{th}$ orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of $n$ is ($hc=1242$ eV nm)

For photo-electric effect with incident photon wavelength $\lambda$, the stopping potential is $V_0$. Identify the correct variation(s) of $V_0$ with $\lambda$ and $1/\lambda$.