26 questions from the UPSC 2021 examination.
26 questions
Find the minimum magnetic field needed for the Zeeman effect to be observed in a spectral line of 400 nm wavelength when a spectrometer whose resolution is 0.010 nm is used. Write the answer in the nearest high integer.
An $n-p-n$ transistor with $\beta = 49$ is used in common-emitter amplifier mode with $V_{cc} = 10\text{V}$ and $R_L = 2\text{ k}\Omega$. If a $100\text{ k}\Omega$ resistor is connected between the collector and the base of the transistor, calculate the quiescent collector current. Assume $V_{BE} = 0$.
The quantum numbers of two electrons in a two valence electron atom are; $n_1 = 8 \quad l_1 = 4 \quad s_1 = \frac{1}{2}$ $n_2 = 7 \quad l_2 = 2 \quad s_2 = \frac{1}{2}$ (i) Assuming $L-S$ coupling, find the possible value of $L$ and hence of $J$. (ii) Assuming $j-j$ coupling, find the possible values of $J$.
A particle is moving in a one dimensional box of width $50\mathring{\text{A}}$ and infinite height. Calculate the probability of finding the particle within an interval of $15\mathring{\text{A}}$ at the centres of the box when it is in its state of least energy.
Calculate the probability of finding a simple harmonic oscillator within the classical limits if the oscillator is in its normal state. Also show that if the oscillator is in its normal state, then the probability of finding the particle outside the classical limits is approximately 16%.
A solid contains a dilute concentration of $Nd^{3+}$ ions, each of which possess three $4f$ electrons. Assuming that there are $10^{25}\text{ m}^{-3}$ of these ions, calculate the magnetic susceptibility of the sample at $1\text{ K}$.
Calculate the pinch-off voltage for $n$-channel silicon FET with a channel width of $6 \times 10^{-4}\text{ cm}$ and a donor concentration of $10^{15}\text{ cm}^{-3}$. Given that dielectric constant of silicon is 12.
Calculate the Larmor precessional frequency for a magnetic induction field of 0.5 T. Hence calculate the splitting in wave numbers of a spectral line due to normal Zeeman effect for the same field.
What are the properties of the particles made up of the following quarks ? (a) $u\bar{d}$ (b) $\bar{u}d$ (c) $dds$ (d) $uss$
The first line in the pure rotational spectrum of HCl appears at $21.18\text{ cm}^{-1}$. Calculate bond length of the molecule. Given atomic masses of H and Cl are 1.008 and 35.45 amu, respectively.
Show that in the nuclear shell model, the level spacing between major oscillator shells is approximately $\hbar \omega = 41 A^{-1/3}\text{ MeV}$.
What are chain reactions ? What do you mean by critical size of the core in which chain reaction takes place ?
Explain the phenomenon of internal conversion and define the internal conversion coefficient. Discuss under what conditions the internal conversion process becomes important.
Explain spin-orbit coupling. Discuss the splitting of spectral lines of H-atom due to spin-orbit coupling.
A beam of 12 eV electron is incident on a potential barrier of height 25 eV and width 0.05 nm. Calculate the transmission coefficient.
Sketch the dc load line for the circuit shown.
Normalised wave function of hydrogen atom for 1s state is $\psi_{100} = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}$, where $a_0 = \frac{h^2}{me^2}$ being the Bohr radius. Calculate the expectation value of potential energy in this state.
What is the importance of study of deuteron ? Obtain the solution of Schrödinger equation for ground state of deuteron and show that deuteron is a loosely bound system.
A particle of rest mass $m_0$ has a kinetic energy $K$, show that its de Broglie wavelength is given by $\lambda = \frac{hc}{\sqrt{[K(K + 2m_0 c^2)]}}$ Hence calculate the wavelength of an electron of kinetic energy 2 MeV. What will be the value of $\lambda$ if $K \ll m_0 c^2$?
Using Pauli spin matrices prove that, (i) $\sigma_x \sigma_y + \sigma_y \sigma_x = 0; \sigma_y \sigma_z + \sigma_z \sigma_y = 0; \sigma_z \sigma_x + \sigma_x \sigma_z = 0$ (ii) $\sigma_+ \sigma_- = 2(1 + \sigma_z)$ (iii) $\sigma_a + \sigma_b = i \sigma_y$, where $a \neq \beta \neq \gamma$
$p^0$ and $K^0$ mesons both decay mostly to $\pi^+$ and $\pi^-$. Explain why the mean lifetime of $p^0$ is shorter ($\sim 10^{-23}\text{s}$) compared to the mean lifetime of $K^0$ ($\sim 10^{-10}\text{s}$).
In the metallic state the transition metal scandium has a single electron in $3d$ subshell. Calculate the values of total angular momentum $J$ and the Lande splitting factor $g$ and use these values to determine the energy of the lowest energy dipole moment in a field of $0.5\text{ T}$.
Using the expression for internal energy $U = 3N \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1}$, show that Einstein specific heat capacity is given by; $C = 3R \left(\frac{\hbar \omega}{k_B T}\right)^2 \frac{e^{\hbar \omega / k_B T}}{\left(e^{\hbar \omega / k_B T} - 1\right)^2}$ Also show that Einstein specific heat capacity given above is proportional to $e^{-\hbar \omega / k_B T}$ at very low temperature.
In observing the Raman spectrum of a sample using $3637\mathring{\text{A}}$ as the exciting line, one gets stoke line at $3980\mathring{\text{A}}$. Deduce the Raman shift in $\text{m}^{-1}$ units. Compute the wavelength in $\mathring{\text{A}}$ for corresponding stokes and antistokes lines if the exciting line is $6465\mathring{\text{A}}$.
Find the uncertainty in the momentum of a particle when its position is determined within 0.02 cm. Find also the uncertainty in the velocity of an electron and $\alpha$-particle respectively when they are located within $15 \times 10^{-8}\text{ cm}$.
An X-ray beam of wavelength ($\lambda_1$) undergoes a first order Bragg reflection at a Bragg angle of $30^{\circ}$. X-ray of wavelength 97 nm undergoes 3rd order reflection at a Bragg angle of $60^{\circ}$. Consider that the two beams are reflected from the same set of planes. Find the value of $\lambda_1$.