Dual Nature of Radiation and Matter

Consider a hydrogen atom with $v_k,r_k$, and $K_k$ denoting the velocity, orbital radius and kinetic energy of the electron in the $k^{\text{th}}$ orbit, respectively. The electron undergoes a transition from the $n^{\text{th}}$ orbit, emitting radiation corresponding to the Lyman series. Considering $h$ to be the Planck's constant and $\epsilon_0$ the permittivity of the free space, the correct statement(s) is/are:

For a certain metal, when monochromatic light of wavelength $\lambda$ is incident, the stopping potential for photoelectrons is $3V_0$. When the same metal is illuminated by light of wavelength $2\lambda$, then the stopping potential becomes $V_0$. The threshold wavelength for photoelectric emission for the given metal is $\alpha\lambda$. The value of $\alpha$ is ________.

The graph shows variation of stopping potential $V_0$ with the frequency $\nu$ of the incident radiation for three photosensitive metals $X_1$, $X_2$ and $X_3$. Which metal will give out electrons with greater kinetic energy, for the same wavelength of incident radiation?

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

The de Broglie wavelength associated with an electron accelerated through a potential difference V is $\lambda_e$ and the de Broglie wavelength associated with a proton accelerated through the same potential difference is $\lambda_p$. If their corresponding masses are $m_e$ and $m_p$, respectively, then the ratio of their de Broglie wavelengths $\left(\dfrac{\lambda_e}{\lambda_p}\right)$ is ________.

An electron of mass $m$ is moving in an electric field $\vec E=-2E_0\hat i$ ($E_0$ = constant > 0), with an initial velocity $\vec V=v_0\hat i$ ($v_0$ = constant > 0). If $\lambda_0=\dfrac h{4mv_0}$, its de Broglie wavelength at time $t$ is ________.
($e$ = charge of electron)

Light source having wavelength 331 nm is used to generate photo-electrons whose stopping potential is 0.2 V. The work function of the used metal in the experiment is $\alpha\times10^{-19}$ J. The value of $\alpha$ is ________.
($h=6.62\times10^{-34}$ J s, $e=1.6\times10^{-19}$ C and $c=3\times10^8$ m/s)

An electron is travelling with a velocity $v$ in free space and when it enters a medium, its velocity is reduced by 20%. The de Broglie wavelength of electron in the medium is $\alpha\lambda_0$, where $\lambda_0$ is its de Broglie wavelength in free space. The value of $\alpha$ is ________.

The ratio of momentum of the photons of the 1st and 2nd line of Balmer series of Hydrogen atoms is $\alpha/\beta$. The possible values of $\alpha$ and $\beta$ are:-

The de Broglie wavelength for an electron accelerated through the potential difference of $V_1$ volt is $\lambda_1$. When the potential difference is changed to $V_2$ volt, the associated de Broglie wavelength is increased by 50%. If $(V_1/V_2)=(9/\alpha)$, then the value of $\alpha$ is ________.

A monochromatic source of light operating at 15 kW emits $2.5\times10^{22}$ photons/s. The region of an electromagnetic spectrum to which the emitted electromagnetic radiation belongs to ________.

(Take $h=6.6\times10^{-34}$ J.s and $c=3\times10^8$ m/s).

$K_1$ and $K_2$ be the maximum kinetic energies of photoelectrons emitted from a surface of a given material for the light of wavelength $\lambda_1$ and $\lambda_2$, respectively. If $\lambda_1=2\lambda_2$ then the work function of material is given by :

A cube of unit volume contains $35\times10^7$ photons of frequency $10^{15}$ Hz. If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is $\alpha\times10^{-9}$ T. Taking permeability of free space $\mu_0 = 4\pi\times10^{-7}$ Tm/A, Planck's constant $h = 6\times10^{-34}$ Js and $\pi = \dfrac{22}{7}$, the value of $\alpha$ is _____.

Consider an electron in the $n = 3$ orbit of a hydrogen-like atom with atomic number $Z$. At absolute temperature $T$, a neutron having thermal energy $k_BT$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $T = \dfrac{Z^2h^2}{\alpha\pi^2a_0^2m_Nk_B}$, (where $h$ is the Planck's constant, $k_B$ is the Boltzmann constant, $m_N$ is the mass of the neutron and $a_0$ is the first Bohr radius of hydrogen atom) then the value of $\alpha$ is _____.

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $\nu_1$ and ejects the electron with a kinetic energy of 10 eV. The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency $\nu_2$. The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV. It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is _____.

A Hydrogen-like atom has atomic number $Z$. Photons emitted in the electronic transitions from level $n = 4$ to level $n = 3$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is 1.95 eV. If the photoelectric threshold wavelength for the target metal is 310 nm, the value of $Z$ is _____.

[Given: $hc = 1240$ eV-nm and $Rhc = 13.6$ eV, where $R$ is the Rydberg constant, $h$ is the Planck's constant and $c$ is the speed of light in vacuum]

Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option.

[Given: Wien's constant as $2.9\times10^{-3}$ m-K and $\dfrac{hc}{e} = 1.24\times10^{-6}$ V-m]

When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0\ \text{V}$. This potential drops to $0.6\ \text{V}$ if another source with wavelength four times that of the first one and intensity half of the first one is used. What are the wavelength of the first source and the work function of the metal, respectively? [Take $\frac{hc}{e} = 1.24 \times 10^{-6}\ \text{J m C}^{-1}$.]

When light of frequency twice the threshold frequency is incident on the metal plate, the maximum velocity of emitted electron is $v_1$. When the frequency of incident radiation is increased to five times the threshold value, the maximum velocity of emitted electron becomes $v_2$. If $v_2=x\,v_1$, the value of $x$ will be ______.

The light of two different frequencies whose photons have energies 3.8 eV and 1.4 eV respectively, illuminate a metallic surface whose work function is 0.6 eV successively. The ratio of maximum speeds of emitted electrons for the two frequencies respectivly will be :