UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Construct the Hamiltonian of a charged particle with charge q and mass m moving with the velocity \vec{v} in the external electromagnetic field, \vec{E}=E_0 \hat{i}, \vec{B}=B_0 \hat{k}, where E_0 and B_0 are constants.

Using the set of four Maxwell's equations, obtain the Lorentz condition relation between the scalar potential \phi and vector potential A, i.e., \nabla \cdot A + \frac{1}{c^2} \frac{\partial \phi}{\partial t} = 0 and discuss the gauge transformations, Lorentz gauge and Coulomb gauge.

A parallel plate capacitor has plate area =4.0\ \mathrm{cm^2} and plate separation =2.0\ \mathrm{mm}. An a.c. voltage V=20\sin(5\times10^{3}t) volts is applied across the plates. If the dielectric constant of the medium between the plates is \varepsilon_r=2.0, calculate the displacement current.

What are the limitations of Rayleigh-Jeans law in explaining the spectrum of radiations from a blackbody? Explain how these limitations were overcome in Planck's radiation law. Deduce Wien's displacement law from Planck's radiation law.

A current carrying circular wire loop of radius 1.0 cm has a magnetic moment 2.0\,\mathrm{mJ/T}. Determine the magnetic field at an axial distance of 3.0 cm from the centre of the loop.

A 12.0 V battery is connected at t=0 to a series combination of a resistor R=10.0\Omega and an inductor L=5.0\mathrm{H}. At what rate is energy being stored in the inductor when the current in the circuit is 0.4 A?

Two solenoids have 500 and 800 turns of wire and are placed co-axially close to each other. A current of 5.0 A in the first solenoid produces an average flux of 200\mu\mathrm{Wb} through its each turn and a flux of 100\mu\mathrm{Wb} through each turn of the second solenoid. Find the self-inductance of the first solenoid and the mutual inductance of the solenoids.

A uniformly magnetized sphere of radius R has magnetization \vec{M}=M_0\hat{z}. If the scalar magnetic potentials inside and outside the sphere are given as under \phi_m=\frac{M_0}{3}z;\ r\leq R and \phi_m=\frac{M_0R^3}{3r^2}\cos\theta;\ r>R where, r,\theta are two spherical coordinates, find the magnetic field inside and outside the sphere.

Define a plane electromagnetic wave. A plane polarized wave is incident on the interface between two dielectric media. Obtain expressions for the amplitudes of the reflected and transmitted waves when the incident wave is polarized with its electric field B vector perpendicular to the plane of incidence. Discuss the phase relationships of the reflected and transmitted waves with respect to the incident wave.

A 0.5\,\mathrm{m} long cylindrical medium between two conducting plates has uniform charge density of 100\,\mathrm{nC/m^3}. The axis of the cylindrical medium is along z-axis. The left plate is at z=0 and has a potential of 10\,\mathrm{kV} and the right plate is grounded. Determine the electric field at axial distance z=0.2\,\mathrm{m}.

Write down Maxwell's equations in integral form. Explain the significance of each of these equations.

(i) How are linearly polarised and circularly polarised lights generated from unpolarised light and are analysed?

(ii) A calcite crystal is to be used for producing a quarter wave plate. The refractive indices of the crystal for ordinary ray and extraordinary ray are 1\cdot 65836 and 1\cdot 48641 respectively. What is the thickness of the plate required if the light of wavelength 632\cdot 8\text{ nm} is to be used?

(i) When a monochromatic light passes through a single slit, dark and bright fringes are observed on a screen kept far away from the slit. Draw an amplitude and intensity pattern of the light on the screen.

(ii) Write an expression for intensity distribution and give the conditions for bright and dark fringes.

(iii) If the slit width is 0\cdot 3\text{ mm} and the distance between the screen and the slit is about 30\text{ cm}, then what is the angle of diffraction for the first minimum? (\lambda = 5 \times 10^{-5}\text{ cm})

(i) Give the equation of motions for a particle executing simple harmonic motion for undamped, damped and forced vibrations, and explain the terms.

(ii) Give a plot between the displacement and time for each case of the above and interpret.

(iii) What are the conditions for critical damping and resonance?

Calculate the focal lengths of the components of an achromatic telescope objective having a power of +3\cdot 333 diopter, made from crown glass and flint glass, whose dispersive powers are 0\cdot 012 and 0\cdot 02 respectively.

With the help of a neat diagram, explain spherical aberration. Briefly discuss the methods to minimize spherical aberration.

Explain the principle and working of He-Ne laser. What is the role of He gas? Why is it necessary to use narrow tube? How many longitudinal modes can be excited for an He-Ne laser in a cavity of length 30 cm and having half width of gain profile of laser material 2 \times 10^{-3}\,\mathrm{nm}? The emission wavelength is 6328\,\mathring{\mathrm{A}}.

Distinguish between positive and negative crystals in terms of double refraction. How are these crystals used to make quarter wave plates? Explain how the quarter wave plate is used in producing elliptically and circularly polarized light.

When the two waves of nearly equal frequencies interfere, then show that the number of beats produced per second is equal to the difference of their frequencies.

What do you mean by spherical aberration of a lens? Show that if two plano-convex lenses are kept at a distance equal to the difference of their focal lengths, the spherical aberration would be minimum.