Browse & Filter

Consider an isolated system of ideal gas of $N$ molecules
contained in a volume $V$ and having an energy
$$E=\sum_{i=1}^{3N}\frac{p_i^2}{2m}.$$ Show that the number of states in the
energy range $U-\Delta$ and $U$ is
of the system is given by\hfill [5]
\begin{eqnarray*}
\frac{1}{\Gamma(3N/2+1)}\frac{3N\Delta}{2U} \Big(\frac{mU V^{2/3}
}{2\pi\hbar^2}\Big)^{3N/2}
\end{eqnarray*}

The internal energy $U$ (of a single component thermodynamic system) expressed as a function of entropy $S$, and volume $V$, is of the form $$U(S,V)=a\ S^{4/3}V^\alpha,$$ where $a$ and $\alpha$ are 

1. What is the value\footnote{HINT : $U,\ S,\ {\rm and}\ V$ are extensive thermodynamic properties. Therefore $U$ is a first order homogeneous function of $S$, and $V$. In other words $U(\lambda S,\lambda V)=\lambda U(S,V).$} of $\alpha$ ?
2. What is the temperature of the system ?
3. What is the pressure of the system ?
4. The pressure of the system obeys a relation given by $$P=\omega U/V,$$ where $\omega$ is a constant. Find the value of $\omega$.
5. if the energy of the system is held constant, the pressure and volume are related by $$PV^\gamma={\rm constant}.$$ Find $\gamma$.

Palash Pal

Use Gauss's law to find the electric field inside a uniformly charged
solid sphere of radius \(R\) and carrying charged density $\rho$. State facts
other than Gauss's law which you might have used in your answer.

A solid sphere of radius \(R\) carries a charge density \(\rho(\vec{r})\).
Show that the average of the electric field inside the sphere is
\[\vec{E}= - \frac{1}{4\pi\epsilon_0} \frac{\vec{p}}{R^3},\]
where \(\vec{p}\) is the total dipole moment of the sphere.

A plane carries a uniform charge density $\sigma_0$ per unit area. A
central circular hole is cut removing the charge in the circular disk.
Find the electric field above the center of the hole at a distance $d$
from the center.

In a monoatomic crystalline solid each atom can occupy either
a regular lattice site or an interstitial site. The energy of an atom at an
interstitial site exceeds the energy of a atom at a lattice site by an amount
\(\epsilon\). Assume that the number interstitial sites equals the number of
lattice sites, and also equals the number of atoms \(N\). Calculate the entropy
of the crystal in the state where exactly \(n\) of the atoms are at the
interstitial sites. What is the temperature of the crystal in this state,if the
crystal is in thermal equilibrium? If \(\epsilon=1 \text{ev}\) and the
temperature of the crystal is 300K, what is the fraction of the atoms at the
interstitial sites?

Show that the number of photons in a cavity at temperature $T$ and having a unit volume is
$$ N\,=\, j \left(\frac{kT}{\hbar c}\right)^3 $$
where $j$ is a numerical constant.

Use the above result to show that the specific heat of a photon gas in proportional to $T^3$

The average kinetic energy of the hydrogen atoms in a certain stellar atmosphere ( assumed to be in equilibrium) is $1$ electron volt.

(a) What is the temperature in Kelvin?

(b) What is the ratio of the number of atoms in the $N=3$ state to the number in the ground state.

(c) Discuss qualitatively the ration of the number of ionized atoms to the atoms in the $N\,=\,3$ state, with out taking the density of states for the ionized atoms, i.e., taking only the Boltzmann factor. ( Taking the density of states has an interesting consequence, first pointed out by Prof. M.N.Saha . Will discuss
this in the class)

• Using \( e(\nu, T) d\nu =\frac{2\pi h}{c^2})\frac{\nu^3}{e^{\beta h \nu}-1}\, d\nu\), where \(e(\nu, T)\) is called the black-body emissivity, show that the energy radiated per unit area and time in the range \(d\lambda\) of \(\lambda\) (where \(\lambda = c/\nu \) is the wavelength) is \[ \left(\frac{2\pi c^2h}{\lambda^5}\right)(e^{\frac{\beta h c }{\lambda}}-1)^{-1} d\lambda \equiv e(\lambda, T) d\lambda.\]
• Show that the wavelength for which \(e(\lambda, T)\) is a maximum is given by \[ \beta h c = 4.965 \lambda_\text{max}\] What does \(\frac{\lambda_\text{max}\nu_{max}}{c}\) equal?
• Solar radiation has a maximum intensity near \(\lambda = 5\times 10^{-5}\)cm. Assuming that the the sun's surface is in thermal equilibrium, determine its temperature.