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In a Young double slit experiment, the first bright maximum is displaced by y=2\,\text{cm} from the central maximum. If the spacing between slits and distance from the screen are 0.1\,\text{mm} and 1\,\text{m} respectively, find the wavelength of light.
If the system matrix for a thin lens of focal length f is given by : S = \begin{bmatrix} 1 & -1/f \\ 0 & 1 \end{bmatrix} , show that the system matrix for a combination of two thin lenses of focal lengths f_1 and f_2 separated by a distance d can be obtained as S_{12} = \begin{bmatrix} 1 - d/f_2 & -(1/f_1 + 1/f_2 - d/f_1 f_2) \\ d & 1 - d/f_1 \end{bmatrix}
Write down the mathematical representation of Fermat's principle and explain all the notations used. With the help of a neat diagram, show that this principle can be used to obtain the law of refraction : n_1 \sin \theta_1 = n_2 \sin \theta_2 , where n_1 and n_2 are the refractive indices of the two media while \theta_1 and \theta_2 are the angles of incidence and refraction of the light beam.
Show that the group velocity is equal to particle velocity. Also prove that the group velocity of the photons is equal to c, the velocity of light.
In a certain engine, a piston undergoes vertical SHM with an amplitude of 10\,\mathrm{cm}. A washer rests on the top of the piston. As the motor is slowly speeded up, at what frequency will the washer no longer stay in contact with the piston?
State Hamilton's principle for a system of particles. If I and L represent the action integral and the Lagrangian function, respectively, write down the mathematical form of Hamilton's principle and explain clearly the significance of the same.
Using the concept of D'Alembert's principle, show that the generalized force can be defined as Q_j = \sum_i \bar{F}_i \cdot \frac{\partial \bar{r}_i}{\partial q_j} , where \bar{r}_i is the Cartesian coordinate of the i^{\text{th}} particle experiencing external force \bar{F}_i and q_j stands for the generalized coordinate. Discuss the significance of the above expression.
Derive the law of addition of relativistic velocities. Use it to prove that under the Lorentz transformation no two velocities can add upto more than the value of the speed of light.
A particle of rest mass M moving at a velocity u collides with a stationary particle of rest mass m. If the particles stick together, show that the speed of the composite ball is equal to u\alpha M/(\alpha M + m), where \alpha = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} .
Deduce the minimum energy of a gamma ray photon (in MeV), which can cause electron-positron pair production.
A mirror is moving through vacuum with a relativistic speed v in the x-direction. A beam of light with frequency \omega_i is normally incident (from x=\infty) on the mirror.
A sphere of radius R moves with velocity \vec{u} in an incompressible, non-viscous ideal fluid. Calculate the pressure distribution over the surface of the sphere. Do you think that a force is necessary to keep the sphere in uniform motion?
A charged particle is moving under the influence of a point nucleus. Show that the orbit of the particle is an ellipse. Find out the time period of the motion.
The density inside a solid sphere of radius a is given by \rho = \frac{\rho_0 a}{r}, where \rho_0 is the density at the surface and r denotes the distance from the centre. Find the gravitational field due to this sphere at a distance 2a from its centre.
Discuss the problem of scattering of charged particle by a coulomb field. Hence, obtain an expression for Rutherford scattering cross-section. What is the importance of the above expression?
Consider a rigid body rotating about an axis passing through a fixed point in the body with an angular velocity \vec{\omega}. Determine the kinetic energy of such a rotating body in a coordinate system of principal axis. If the earth suddenly stops rotating, what will happen to the rotational kinetic energy? Comment in detail.
If I' and I be the Moments of Inertia of a body about an axis passing through an arbitrary origin and about a parallel axis through the centre of mass respectively, show that I' = MR^2 + I, where \vec{R} is the position vector of the centre of mass with respect to the arbitrary origin and M is the mass of the body.
A body turns a fixed point. Show that the angle between its angular velocity vector and its angular momentum vector about a fixed point is always acute.
A perfect conductor and a superconductor with T_c \sim 10\ \mathrm{K} are subjected to the following conditions:
(i) cooled under applied magnetic field to 4\ \mathrm{K}
(ii) cooled to 4\ \mathrm{K} and then magnetic field is applied with schematic diagrams, explain path of magnetic field lines in all these situations.