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The large distance expansion of potential due to a localized charge distribution is obtained. The first three terms receiving contributions from the monopole, the dipole moment and the quadrupole moment are explicitly displayed.  Important properties of dipole and quadrupole moment are discussed.

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$  $\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$
$\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$  $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$

The field tensor for the electromagnetic fields is defined in terms of the vector potential. It components are expressed in terms of the electric and magnetic fields. The Maxwell's equations are written down in relativistic notation.

A box of volume $2V$ is divided into equal halves by a thin partition. The left side contains perfect gas at pressure $p_L$ and the right side is vacuum. A small hole of area $A$ is punched in the partition at time $t\,=\,0$. What is the pressure in the left had side $p_L(t)$ after a time t ?. Assume the temperature to be constant on both the sides as $T$. Assume Maxwell- Boltzmann statistics.

Using the grand canonical ensemble method, the Bose Einstein and Fermi Dirac distribution formula are derived. An expression for the partition function for photons is proved and result on the mean number of particles in terms of grand canonical partition function is obtained

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Starting from the time dependent Schrodinger equation, it is proved that the average value a dynamical variable \(\hat F\)obeys the equation\begin{equation} {d\over dt}\, \langle \hat{F} \rangle = \,\langle{\partial\over \partial t} \hat{F} \rangle + {1\over i\hbar} \langle\, [\hat{F},\hat{H} ]\, \rangle.\end{equation}

An ideal gas is enclosed in a cylindrical volume with a movable piston. It can be taken from a state 1 to state 2 by several processes as shown in \(P-V \) diagram below. It is given that if the walls are adiabatic the system obeys the equation \(P^3 V^5=\text{const}\) The energy of the state in state 1 is given to be 100J. The pressures and volume at for the two states are given as follows. \[P_1=10^5 \,\text{Pa}, V_1=10^{-3}\, \text{m}^3; P_2 =\Big(\frac{10^5}{32}\Big) \,\text{Pa}, V_2= b\times 10^{-3}\, \text{m}^3.\]  {\textwidth}{ Several paths taking the system from state 1 to state 2 are shown in figure below.

• AB followed by BC
• AD followed by DC
• Linear path connecting A and C
• Some other path AQC

Your are required to compute the internal energy of the system in state 2 by computing work done by selecting a path from one of the above options. If you choose option (d), some other path, specify the path. Compute the work done along the path chosen and find the internal energy of system in state 2.

When a dipole is placed in a uniform electric field, it experiences a torque  given by \(\vec \tau= \vec p \times \vec E\).