Browse & Filter

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qm-lec-16010

Using time dependent Maxwell's equations and considering a charge distribution moving under influence of the electric and magnetic fields, an equation for rate of change of mechanical work done on the charges is derived, see EQ12. This equation is local conservation law of energy. It says that the rate of the sum of change energy of e.m. fields and work done in a volume \(V\) equals to the flow of energy through the boundary of the volume \(V\). The flow though the boundary is given by the Poynting vector \(\vec{S}\) defined in EQ10.

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Consider a paramagnetic system, with variables magnetization $M$, the magnetic field $B$ and absolute temperature $T$. ( We assume it's dependence on pressure as negligible). The equation of state is ( which will be obtained from statistical mechanics later in the course) is
$$ M\,=\,C\frac{B}{T}, $$
where $C$ is a constant ( referred to as the Curie constant, who had experimentally obtained this relation.
The system's internal energy is ( for a one-dimensional system)
$$ U\,=\,-MB.$$
The work done on the system by external surrounding is $-MdB$

(a) Write the expression for $DQ$ in terms of $dM$ and $dB$

(b) Write the equation for entropy change $dS$ in terms of $dM$ and $dB$

(c) Obtain the entropy $S$

Statistical mechanics is based on the fundamental assumption that all microstates of an isolated system are equally

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A tabular comparison of  coordinate and momentum representations is presented.

Show from calculating
$$ \frac{\Delta S}{\Delta E}\,=\,\frac{1}{T}\,=\,k\beta$$
for the case of Fermi-Dirac distribution,
$$ \Omega_{FD}\,=\,\Pi_{i=1}^\infty\left({}^{g_i}C_{N_i}\right) $$
[ notation as used in the class ]. Also for the case of equilibrium we have
$$ N_i\,=\,\frac{g_i}{e^{\alpha+\beta\epsilon_i}\,+\,1} $$
Hint: Consider changes only in two levels say with energies $\epsilon _1$ and $\epsilon_2$. Then argue that the result so obtained is independent of the choice of levels]

Electrodynamics                                                  April 3, 2022
                                            Quiz-I

 Two grounded infinite conducting planes are kept along the XZ and Y Z planes, see Fig.2. A charge q is placed at (4,3) find the force acting on the charge q.

An electron moving with speed of $5.0\times 10^8$cm/sec is shot parallel to an electric field strength of $1.0\times 10^3 $nt/coul arranged so as to retard its motion.

• How far will the electron travel in the field before coming (momentarily) to rest ?
• how much time will elapse?
• If the electric field ends abruptly after $0.8$ cm, what fraction of its initial energy will the electron loose in traversing the field?