List of practice Questions

An RC circuit is connected to two dc power supplies, as shown in the figure. With switch \( S \) open, the capacitor is fully charged. \( S \) is then closed at time \( t = 0 \). The voltage across the capacitor at \( t = 2.4 \, \text{ms} \) is ................ V (Round off to one decimal place). 

Consider a hemispherical glass lens (refractive index is 1.5) having radius of curvature \( R = 12 \, \text{cm} \) for the curved surface. An incoming ray, parallel to the optical axis, is incident on the curved surface at a height \( h = 1 \, \text{cm} \) above the optical axis, as shown in the figure. The distance \( d \) (from the flat surface of the lens) at which the ray crosses the optical axis is ............ cm (Round off to two decimal places). 

A laser beam shines along a block of transparent material of length 2.5 m. Part of the beam goes to the detector \( D_1 \) while the other part travels through the block and then hits the detector \( D_2 \). The time delay between the arrivals of the two light beams is inferred to be 6.25 ns. The speed of light \( c = 3 \times 10^8 \, \text{m/s} \). The refractive index of the block is ............ (Round off to two decimal places). 

The following Zener diode voltage regulator circuit is used to obtain 20 V regulated output at load resistance \( R_L \) from a 35 V dc power supply. Zener diodes are rated at 5W and 10V. The value of the resistance \( R \) is ......... \( \Omega \). 

A particle of mass \( m \) is in an infinite square well potential of length \( L \). It is in a superposed state of the first two energy eigenstates, as given by \( \psi(x) = \frac{2}{\sqrt{3L}} \psi_1(x) + \frac{2}{\sqrt{3L}} \psi_2(x) \). Identify the correct statement(s). \( h \) is Planck’s constant.

The figure shows the cross-section of a hollow cylindrical tank, 2.2 m in diameter, which is half filled with water (refractive index of 1.33). The space above the water is filled with a gas of unknown refractive index. A small laser moves along the bottom surface and aims a light beam towards the center (see figure). When the laser moves a distance of \( S = 1.09\,\text{m} \) or beyond from the lowest point in the water, no light enters the gas. Identify the correct statement(s). (Speed of light = \( 3 \times 10^8\,\text{m/s} \)) 

A time independent conservative force \( \vec{F} \) has the form, \( \vec{F} = 3y\hat{i} + f(x, y)\hat{j} \). Its magnitude at \( x = y = 0 \) is 8. The allowed form(s) of \( f(x, y) \) is (are):

For the given circuit, identify the correct statement(s). 

A beam of light traveling horizontally consists of an unpolarized component with intensity \( I_0 \) and a polarized component with intensity \( I_p \). The plane of polarization is oriented at an angle \( \theta \) with respect to the vertical. The figure shows the total intensity \( I_{\text{total}} \) after the light passes through a polarizer as a function of the angle \( \alpha \), that the axis of the polarizer makes with respect to the vertical. Identify the correct statement(s). 

A thin circular disc lying in the \( xy \)-plane has a surface mass density \[ \sigma(r) = \begin{cases} \sigma_0\left(1 - \frac{r^2}{R^2}\right), & \text{if } r \le R \\ 0, & \text{if } r > R \end{cases} \] where \( r \) is the distance from its center. Its moment of inertia about the z-axis passing through its center is:

A mass \( m \) is connected to a massless spring of spring constant \( k \), which is fixed to a wall. Another mass \( 2m \), having kinetic energy \( E \), collides collinearly with the mass \( m \) completely inelastically (see figure). The entire setup is placed on a frictionless floor. The maximum compression of the spring is: 

A semiconductor \( pn \) junction at thermal equilibrium has the space charge density \( \rho(x) \) profile as shown in the figure. The figure that best depicts the variation of the electric field \( E \) with \( x \) is ( \( W \) denotes the width of the depletion layer): 

The condition for maxima in the interference of two waves \( A e^{i(\frac{k}{\sqrt{2}}(\sqrt{3}x + y) - \omega t)} \) and \( A e^{i(\frac{k}{\sqrt{2}}(x + \sqrt{3}y) - \omega t)} \) is given in terms of the wavelength \( \lambda \) and \( m \), an integer, by:
The electric field of an electromagnetic wave has the form \( \vec{E} = E_0 \cos(\omega t - kz)\hat{i} \). At \( z = 0 \), a test particle of charge \( q \) and velocity \( \vec{v} = 0.5c \hat{k} \) (where \( c \) is the speed of light) is placed. The total instantaneous force on the particle is:

Four charges are placed very close to each other, as shown. The separation between the two charges on the y-axis is \( a \). The separation between the two charges on the x-axis is also \( a \). The leading order (non-vanishing) form of the electrostatic potential, at point \( P \), at a distance \( r \) from the origin (\( r \gg a \)), is: 

The total charge contained within the cube (see figure), in which the electric field is given by \( \vec{E} = K(4x^2 \hat{i} + 3y \hat{j}) \), where \( \varepsilon_0 \) is the permittivity of free space, is: 

For the given circuit, the output \( Y \) is: 

In the figure below, point A is the object and point B is the image formed by the lens. Let \( l_1, l_2 \) and \( l_3 \) denote the optical path lengths of the three rays 1, 2 and 3, respectively. Identify the correct statement. 

Let \((x, y)\) denote the coordinates in a rectangular Cartesian coordinate system \( C \). Let \((x', y')\) denote the coordinates in another coordinate system \( C' \) defined by
\[ x' = 2x + 3y, \quad y' = -3x + 4y \] The area element in \( C' \) is:

For the given circuit, \( V_D \) is the threshold voltage of the diode. The graph that best depicts the variation of \( V_o \) with \( V_i \) is: 

For a semiconductor material, the conventional flat band energy diagram is shown in the figure. The variables Y, X respectively are: 

A current \( I \) is uniformly distributed across a long straight non-magnetic wire (\( \mu_r = 1 \)) of circular cross-section with radius \( a \). Two points \( P \) and \( Q \) are at distances \( \frac{a}{3} \) and \( 9a \), respectively, from the axis of the wire. The ratio of the magnetic fields at points \( P \) and \( Q \) is ................
A particle \( A \) of mass \( m \) is moving with a velocity \( v_A \), and collides elastically with a particle \( B \), of mass \( 2m \). Particle \( B \) is initially at rest. After collision, \( A \) moves with a velocity \( v_A' \). If \( v_B \) is the final speed of \( B \), then \( v_A'^2 = k v_B^2 \). The value of \( k \) is ...............
In an X-ray diffraction experiment with Cu crystals having lattice parameter \( 3.61 \, \text{Å} \), X-rays of wavelength of \( 0.090 \, \text{nm} \) are incident on the family of planes \( \{ 1 1 0 \} \). The highest order present in the diffraction pattern is .......
The wavelength of characteristic K\(_\alpha\) X-ray photons from Mo (atomic number 42) is ............. Å. (Round off to one decimal place).