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 A mass $m$ of a liquid at a temperature $T_1$ is mixed with an
equal mass of the same liquid at a temperature $T_2$. The system is
thermally isolated. Show that the entropy change is given by
$$
2m\,C_p\,\ln\left({(T_1+T_2)/2\over\sqrt{T_1T_2}}\right)
$$
and prove that this is necessarily positive.

Electrodynamics                                                Apr 19, 2019
                                     Quiz-VIII

Question 1:
A student was asked to find the electric field inside
a uniformly polarized sphere. The following answer was received.

Sample Answer:


The answer to the above question as given in Griffiths is

\(\vec{E}=-\frac{\vec{P}}{3\epsilon_0}\)

1. Give your comments why is this answer wrong?
2. Give a counter example showing that the conclusion arrived by the student is incorrect.
3. Give steps to derive the correct answer

Question 2:
Consider a parallel plate capacitor with plates having charges \(Q\) and \(-Q\). An insulating medium fills the region between the plates. Assuming that the linear dimensions of the plates are much larger than the separation between the plates, draw a diagram to show

1. the distribution of free charges on the plates.
2. distribution of the bound charges on the surface of the dielectric material.
3. Use Gauss law to obtain \(\vec{D}\). Use your result for \(\vec{D}\) and obtain \(\vec{E}\), the potential difference between the plates and the capacitance.

The flux of electric field is defined and As a simple example, the flux of the electric field due to a point charge at the center of a sphere is explicitly computed. Other cases are briefly mentioned an statement of Gauss law is given.

An expression for the electric stress tensor is derived for a charge distribution in a volume \(V\). The surface integral of the stress  tensor gives the total electric force on the  charge in the volume \(V\).

A system consists of three particles and each particle can exist in five possible states. Find the total number of microstates and the number of microstates that energy level has two particles assuming

• the particles are non-identical
• are identical bosons
• are identical fermions.

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\average}[2]{\langle#1|#2|#1\rangle}$                  
Making use of analogy between optics ad mechanics we motivate the introduction of the Schrodinger equation.
Here Fermi's "Lectures on Quantum Mechanics" has been followed very closely..