Modern Physics — JEE Advanced PYQs

Consider a hydrogen atom with ${\mathrm{v}}_{\mathrm{k}},{\mathrm{r}}_{\mathrm{k}}$ and ${\mathrm{K}}_{\mathrm{k}}$ 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 …

A nuclear reactor starts producing a radioactive nuclide $X$ from $t=0$, at a constant rate of $\alpha$ per second. Each decay of $X$ produces energy $E_0$, which is utilized to heat a liquid of mass $m$ and specific heat $s$. Assuming no heat loss from the liquid …

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 $k_{\mathrm{B}}T$ has the same de Broglie wavelength as that of this electron. If this temperature is given by …

List-I shows various functional dependencies of energy $(E)$ on the atomic number $(Z)$. Energies associated with certain phenomena are given in List-II.
Choose the option that describes the correct match between the entries in List-I to those in List-II.

| List-I | | List- …

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $v_1$ 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 pho …

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $\mathrm{F}(r)=-kr$, corresponding to the potential energy $V(r)=k{r}^2/2$, where $k$ is a positive force constant and $r$ is the radial distance from the origin. According …

A metal target with atomic number $Z=46$ is bombarded with a high energy electron beam. The emission of X-rays from the target is analyzed. The ratio $r$ of the wavelengths of the $K_{\alpha }$-line and the cut-off is found to be $r=2$. If the same electron beam bombards another m …

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. I …

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.
| (P) | …

In a radioactive decay process, the activity is defined as $A=-\frac{\mathrm{d}N}{\mathrm{d}t}$, where $N(t)$ is the number of radioactive nuclei at time $t$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t=0$. At a later time, the activities of …

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${}_7^{16}\mathrm{N}+{}_2^4\mathrm{He}\to {}_1^1\mathrm{H}+{}_8^{19}\mathrm{O}$ in a laboratory frame is $n$ (in MeV). Assume that ${}_7^{16}\mathrm{N}$ is at rest …

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_b^p$ and the binding energy of a neutron be $E_b^n$ in the nu …

In a radioactive decay chain reaction, ${}_{90}^{230}\mathrm{Th}$ nucleus decays into ${}_{84}^{214}\mathrm{Po}$ nucleus. The ratio of the number of $\alpha$ to number of ${\beta }^{-}$particles emitted in this process is ____.

When light of a given wavelength is incident on a metallic surface, the minimum potential needed to stop the emitted photoelectrons is $6.0V$. This potential drops to $0.6V$ if another source with wavelength four times that of the first one and intensity half of the first o …

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 …

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is

Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?
(A) The ratio of the longest wavelength to the shortest wavelength in Balmer series is 9/5
(B) There is an overlap between the wavelength ranges of Balmer and Paschen series.
(C) The waveleng …

A particle of mass m moves in circular orbits with potential energy $\mathrm{V}(\mathrm{r})=Fr$, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by R and its …

In an X-ray tube, electrons emitted from a filament (cathode) carrying current I hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in emission of continuous and characteristic X-rays. If the filament c …

In a radioactive sample, ${}_{19}^{40}\mathrm{K}$ nuclei either decay into stable ${}_{20}^{40}\mathrm{Ca}$ nuclei with decay constant $4.5\times 10^{-10}$ per year or into stable ${}_{18}^{40}\mathrm{Ar}$ nuclei with decay constant $0.5\times 10^{-10}$ per year. Given that in th …

A free hydrogen atom after absorbing a photon of wavelength ${\lambda }_{\mathrm{a}}$ gets excited from the state $\mathrm{n}=1$ to the state $\mathrm{n}=4$. Immediately after that the electron jumps to $\mathrm{n}=\mathrm{m}$ state by emitting a photon of wavelength …

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency $\Omega$ such that $\frac{4\pi \mathrm{M}\Omega }{\mathrm{h}}=10^{24}{\mathrm{m}}^{-2}$ with h as Planck's constant. N photons of wavelength …

Suppose a ${}_{88}^{226}\mathrm{Ra}$ nucleus at rest and in ground state undergoes $\alpha$-decay to a ${}_{86}^{222}\mathrm{Rn}$ nucleus in its excited state. The kinetic energy of the emitted $\alpha$ particle is found to be 4.44 MeV. ${}_{86}^{222}\mathrm{Rn}$ nuc …

In a radioactive decay chain, ${}_{90}^{232}\mathrm{Th}$ nucleus decays to ${}_{82}^{212}\mathrm{Pb}$ nucleus. Let $N_{\alpha }$ and $N_{\beta }$ be the number of $\alpha$ and ${\beta }^{-}$particles, respectively, emitted in this decay process. Which of the following sta …

In a photoelectric experiment a parallel beam of monochromatic light with power of $200W$ is incident on a perfectly absorbing cathode of work function $6.25eV$. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligi …

Consider a hydrogen-like ionized atom with atomic number $Z$ with a single electron. In the emission spectrum of this atom, the photon emitted in the $n=2$ to $n=1$ transition has energy 74.8 eV higher than the photon emitted in the $n=3$ to $n=2$ transition. The ioniza …

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relatio …

An electron in a hydrogen atom undergoes a transition from an orbit with quantum number $n_i$ to another with quantum number $n_f.V_i$ and $V_f$ are respectively the initial and final potential energies of the electron. If $\frac{V_i}{V_f}=6.25$, then the sm …

${}^{131}$ I is an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with ${}^{131}\mathrm{I}$ is injected into the blood of a person. The activity of the amount of ${}^{131}\mathrm{I}$ injected was …

A photoelectric material having work-function ${\varphi }_0$ is illuminated with light of wavelength $\lambda \left(\lambda <\frac{hc}{{\varphi }_0}\right)$. The fastest photoelectron has a de Broglie wavelength ${\lambda }_d$. A change in wavelength of the incident light by …

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 a …

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge $Ze$ are defined by their principal quantum number $n$, where $n\gg 1$. Which of the following statement(s) is(are) true?
(A) Relative change in the radii of two consecutive orbi …

A hydrogen atom in its ground state is irradiated by light of wavelength 970 Å. Taking $hc/e=1.237\times 10^{-6}$ eV m and the ground state energy of hydrogen atom as -13.6 eV, the number of lines present in the emission spectrum is

The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by

The measured masses of the neutron, ${}_1^1\mathrm{H},{}_7^{15}\mathrm{N}$ and …

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is …

light of wavelength ${\lambda }_{\text{ph }}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\varphi$ and the anode is a wire mesh of conducting material kept at a distance $d$ from the cathode. A potential diff …

The isotope ${}_5^{12}\mathrm{B}$ having a mass 12.014 u undergoes $\beta$-decay to ${}_6^{12}\mathrm{C}$. ${}_6^{12}\mathrm{C}$ has an excited state of the nucleus $({}_6^{12}{\mathrm{C}}^{\ast })$ at 4.041 MeV above its ground state. If ${}_5^{12}\mathrm{B}$ dec …

Consider a hydrogen atom with its electron in the $n^{\text{th}}$ 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 ( $hc=1242\mathrm{eV}\mathrm{nm}$ )

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life $T$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is $12.5\%$ of the electrical power available from the plant a …

For photo-electric effect with incident photon wavelength $\lambda$, the stopping potential is ${\mathrm{V}}_0$. Identify the correct variation(s) of ${\mathrm{V}}_0$ with $\lambda$ and $1/\lambda$.

Match the nuclear processes given in Column I with the appropriate option(s) in Column II.
| (B) | Fission in a nuclear reactor | …

For a radioactive material, its activity $A$ and rate of change of its activity $R$ are defined as $A=-\frac{dN}{dt}$ and $R=-\frac{dA}{dt}$, where $N(t)$ is the number of nuclei at time $t$. Two radioactive sources $P$ (mean life $\tau$) and $Q$ (mean life $2\tau$) have the …

An electron in an excited state of ${\mathrm{Li}}^{2+}$ ion has angular momentum $3h/2\pi$. The de Broglie wavelength of the electron in this state is $p\pi a_0$ (where $a_0$ is the Bohr radius). The value of $p$ is

A fission reaction is given by ${}_{92}^{236}\mathrm{U}{\to }_{54}^{140}\mathrm{Xe}{+}_{38}^{94}\mathrm{Sr}+\mathrm{x}+\mathrm{y}$, where x and y are two particles. Considering ${}_{92}^{236}\mathrm{U}$ to be at rest, the kinetic energies of the products are denoted by …

A metal surface is illuminated by light of two different wavelengths $248nm$ and $310nm$. The maximum speeds of the photoelectrons corresponding to these wavelengths are $u_1$ and $u_2$, respectively. If the ratio $u_1:u_2=2:1$ and $hc=1240eVnm$, the wor …

If ${\lambda }_{Cu}$ is the wavelength of $K_{\alpha }$ X-ray line of copper (atomic number 29) and ${\lambda }_{Mo}$ is the wavelength of the $K_{\alpha }$ X-ray line of molybdenum (atomic number 42), then the ratio ${\lambda }_{Cu}/{\lambda }_{Mo}$ is close to

A pulse of light of duration $100ns$ is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is $3\times 10^{8}m{s}^{-1}$. The final momentum of the object is

The work functions of Silver and Sodium are 4.6 and $2.3eV$, respectively. The ratio of the slope of the stopping potential versus frequency plot for Silver to that of Sodium is

A freshly prepared sample of a radioisotope of half-life $1386s$ has activity $10^3$ disintegrations per second. Given that $\ln 2=0.693$, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first $80s$ after p …

The radius of the orbit of an electron in a Hydrogen-like atom is $4.5a_0$, where $a_0$ is the Bohr radius. Its orbital angular momentum is $\frac{3h}{2\pi }$. It is given that $h$ is Planck constant and $R$ is Rydberg constant. The possible wavelength(s), when th …