List of practice Questions

The angular momentum of a particle relative to the origin varies with time (\(t\)) as \(\vec{L} = (4\hat{i} + \alpha t^2 \hat{j})\, \mathrm{kg \cdot m}^2/\mathrm{s}\), where \(\alpha = 1\, \mathrm{kg \cdot m}^2/\mathrm{s}^3\). The angle between \(\vec{L}\) and the torque acting on the particle becomes \(45^\circ\) after a time of ............ s.
 

An ideal gas undergoes an isothermal expansion along path AB, adiabatic expansion along BC, isobaric compression along CD, isothermal compression along DE, and adiabatic compression along EA, as shown in the figure. The work done by the gas along the process BC is 10 J. The change in the internal energy along process EA is 16 J. The absolute value of the change in the internal energy along the process CD is ............ J. 

The figure shows a circuit containing two diodes \(D_1\) and \(D_2\) with threshold voltages \(V_{TH}\) of 0.7 V and 0.3 V, respectively. Considering the simplified diode model, which assumes diode I–V characteristics as shown in the plot on the right, the current through the resistor \(R\) is ......... µA. 

The line integral of the vector function \(u(x, y) = 2y \, \hat{i} + x \, \hat{j}\) along the straight line from (0, 0) to (2, 4) is ..........
 

Figure shows a circuit diagram comprising Boolean logic gates and the corresponding timing diagrams showing digital signals at various points in the circuit. Which of the following is/are true? 

In the circuit shown in the figure, both OPAMPs are ideal. The output for the circuit \(V_{out}\) is: 

For an unbiased Silicon \(n\text{-}p\text{-}n\) transistor in thermal equilibrium, which one of the following electronic energy band diagrams is correct? (\(E_c\) = conduction band minimum, \(E_v\) = valence band maximum, \(E_F\) = Fermi level.) 

Consider the crystal structure shown in the figure, where black and grey spheres represent atoms of two different elements and \(a\) denotes the lattice constant. The Bravais lattice for this structure is: 

The normalized ground-state wave function of a one-dimensional quantum harmonic oscillator with force constant \(K\) and mass \(m\) is \(\psi_0(x) = \left(\frac{\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2/2}\), where \(\alpha = \frac{m\omega_0}{\hbar}\) and \(\omega_0^2 = \frac{K}{m}\). Which one of the following is the probability of finding the particle outside the classically allowed region?
(The classically allowed region is where the total energy is greater than the potential energy.) 
 

In which one of the following limits does the Fermi-Dirac distribution \(n_F(\epsilon, T) = \left(e^{\frac{\epsilon - \mu}{k_B T}} + 1\right)^{-1}\) and the Bose-Einstein distribution \(n_B(\epsilon, T) = \left(e^{\frac{\epsilon - \mu}{k_B T}} - 1\right)^{-1}\) reduce to the Maxwell-Boltzmann distribution?
 

Consider an annular region in free space containing a uniform magnetic field in the \(z\)-direction, schematically represented by the shaded region in the figure. A particle having charge \(Q\) and mass \(M\) starts from point \(P(a, 0, 0)\) in the \(+x\)-direction with constant speed \(v\). If the radii of inner and outer circles are \(a\) and \(b\), respectively, the minimum magnetic field required so that the particle returns to the inner circle is: 

A beam of unpolarized light of intensity \(I_0\) falls on a system of four identical linear polarizers placed in a line as shown in the figure. The transmission axes of any two successive polarizers make an angle of \(30^\circ\) with each other. If the transmitted light has intensity \(I\), the ratio \(\dfrac{I}{I_0}\) is: 

The volume integral \(\displaystyle \int_V e^{-\left(\frac{r}{R}\right)^2} \vec{\nabla} \cdot \left(\frac{\hat{r}}{r^2}\right) d^3r\), where \(V\) is the volume of a sphere of radius \(R\) centered at the origin, is equal to:
 

Consider the motion of a quantum particle of mass \(m\) and energy \(E\) under the influence of a step potential of height \(V_0\). If \(R\) denotes the reflection coefficient, which one of the following statements is true? 

The figure below shows a cubic unit cell with lattice constant \(a\). The shaded crystallographic plane intersects the x-axis at 0.5a. The Miller indices of the shaded plane are: 

For a system undergoing a first-order phase transition at a temperature \(T_c\), which one of the following graphs best describes the variation of entropy (S) as a function of temperature (T)?