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In this section examples are given to show that the flux rule is not always applicable.

Consider a 2 dimensional phase space ( $q,p$) with a rectangular region defined by four corners as shown.

If the region ABCD is the phase space region at time time t = 0 , find the region $A'B'C'D'$ at time t given the Hamiltonian is
$$ H\,=\,\frac{p^2}{2m}\,-\, m a q $$
and explicitly verify that the area is constant. Take the coordinates of A,B,C and D as $(q_A,p_A)\,,\,(q_B,p_A)\,,\,(q_B,p_C)$ and $(q_A,p_C)$ respectively

 Electrodynamics                                               March 11, 2019

                                      Quiz-IV

In case of a charge distribution having symmetry about \(z\)-axis,

\(Q_{xy}=Q_{yz}=Q_{zx}=0\) and \(Q_{yy}=Q_{xx}\) Also trace
\(Q_{xx}+Q_{yy}+Q_{zz}\) is always zero. Thus the quadrupole moment tensor can
be specified by a single number
\(Q= 2(Q_{zz}-Q_{xx})\). Calculate \(Q\) for six equal charges placed on corners
of a regular hexagon of sides \(a\).

The definition of quadrupole moment tensor components is
\begin{eqnarray}
Q_{xx} &=&\frac{1}{2} \sum_\alpha Q_\alpha (3x^2_\alpha - r^2_\alpha)\\
Q_{yy} &=& \frac{1}{2} \sum_\alpha Q_\alpha (3y^2_\alpha - r^2_\alpha)\\
Q_{zz}&=& \frac{1}{2} \sum_\alpha Q_\alpha (3z^2_\alpha - r^2_\alpha)
\end{eqnarray}
In this case \(Q_{xz}=Q_{yz}=0\) because all charges are in \(XY\) plane
\((z=0)\). Also \(Q_{xy}=0\) due to symmetry reflection symmetry in the \(X,
Y\) axes.
For all the charges \(z=0, r=a\). Hence
\begin{equation*}
\sum_{\alpha=1}^6 z^2=0 \qquad \sum_{\alpha=1}^6 r^2= 6a^2
\end{equation*}
For four charges \(x^2= \frac{3a^2}{4}\) and for two charges \(x=0\). Therefore
\begin{equation*}
\sum_{\alpha=1}^6 3 x^2 = 9a^2
\end{equation*}
Thus we get
\begin{eqnarray}
Q_{xx} &=& \frac{1}{2} \sum_\alpha Q_\alpha (3x^2_\alpha - r^2_\alpha)=
Q (9a^2 - 6a^2) = 3Qa^2 \\
Q_{zz} &=& \frac{1}{2} \sum_\alpha Q_\alpha (3z^2_\alpha - r^2_\alpha)= - 6Qa^2
\end{eqnarray}
and the final answer is
\[Q= 2(-6Qa^2- 3Qa^2) = -18 Qa^2.\]

The Maxwell's equations for electrostatic are derived from Coulomb's law which has been formulated based on experiments. This provides initial experimental evidence for the Maxwell's equations. We discuss two applications of Maxwell's equations. The first result is that  the electric field inside an empty cavity in conductors is proved to be zero. The second result is  an  expression for electric stress tensor is derived. The surface integral of the electric tensor gives the force on charge distribution.

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qm-lec-17001

The vector and scalar potentials are defined in terms of the fields. Using the Maxwell's equations the wave equation for the potentials are derived in the Lorentz gauge.

$$pV\,=\,A(T)\,+\,B(T)p\,+\,C(T)p^2 $$
Find $C_p(T,p)$ in terms of $C_p(T,p_0)$ and $p\,,\,p_0$ ( initial and the final pressures) and $A(T)\,,\,B(T)\,,\,C(T)$ and their derivatives with respect to temperature $T$. [ Write an appropriate expression for
$$\frac{\partial C_p}{\partial p} $$ and use it to obtain $C_p$]

The canonical partition function for an ideal gas is computed and ideal gas equation is derived. A measurement of the Boltzmann constant k is discussed using effusion of gas molecules through a hole. Distribution function of molecules in presence of gravity is as function of height is derived.