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A gold nucleus contains a positive charge equal to that of 79 protons. An $\alpha$ particle, $Z=2$, has kinetic energy $K$ at points far away from the nucleus and is traveling directly towards the  charge, the particle just touches the surface of the charge and is reversed in direction. relate $K$ to the radius of the gold nucleus. Find the numerical value of kinetic energy in MeV is the radius $R$ is given to be $5 \times10^{-15}$ m.

[ 1 MeV = $10^6$ eV and 1 eV = $1.6\times10^{-16}$]

A circular coil is formed from a  wire of length $L$ with $n$ turns. The coil carries a current $I$ and is placed in an external uniform magnetic field  $B$. Show that  maximum torque developed is  $\displaystyle\frac{IBL^2}{4n\pi}$.

A uniform electric field \(E_m\) is set up in a medium of dielectric constant \(\kappa\). Prove that the field inside a spherical cavity in the medium is given by \begin{equation*}   E = \frac{3\kappa E_m}{2\kappa+1} \end{equation*}

The canonical partition function of a system of $N$ hypothetical particles each of mass $m$, confined to a volume $V$ at temperature $T$ is given by, $$Q(T,V,N) = V^N\left(\frac{2\pi k_B T}{m}\right)^{5N/2}.$$ Determine the equation of state of the hypothetical system. Also find $C_V$ - heat capacity at constant volume. Identify the hypothetical system. How many degrees of freedom does each particle of the hypothetical system have ?

KPN

There are three single particle quantum levels : a  non degenerate ground state of energy zero and a  doubly degenerate excited state of energy $\epsilon=10k_B \ln 18$ joules. Non-interacting particles obeying Maxwell-Boltzmann statistics occupy these three quantum levels. The closed system is described by a canonical ensemble at temperature $T$. Find the temperature below which more than $90$ percent of the particles would be found in the ground state.

KPN

Consider N point particles inside a box of volume V and at temperature T. The energy function is given by $$ H\,=\,\frac{p^2}{2M}\,+\,V(|\vec{r}_i\,-\,\vec{r}_j|) $$ where $$ V(r) \,=\,0 \qquad\rm{for}\,r\,>\,a$$ $$ =\,\infty\qquad \rm{when}\, r\,<\,a $$ Calculate the canonical partition function $Z_c$ and show that $$ Z_c\,=\,\frac{1}{N!}\left(\frac{2\pi m k T}{h^2}\right)^{3/2} Q $$ where $$Q\,=\,V^N\left(1\,-\,\frac{4\pi a^3}{3}\frac{N(N+1)}{2V}\right) $$ with correction of order $V^{(N-2)}$. Evaluate the corrections to $PV\,=\,NkT$ to the next non vanishing order.

HSMani

A box of volume, \(V = L^3\) , contains an ideal gas of \(N\) identical atoms, each of which has spin, \(s = 1/2\), and magnetic moment, \(\mu\). A magnetic field, \(B\) is applied to the system. (a) Compute the partition function for this system. (b) Compute the internal energy and the heat capacity. (c) What is the magnetization?

The internal energy density u of a gas is function of T only and further $$ P\,=\,\frac{1}{3}u(T) $$. Find the functional form of $u(T).$

HSMani

Calculate the fluctuation in energy $(\overline{\Delta E})^2\,=\,\overline{E^2}\,-\,\overline{E}^2$ for the grand canonical ensemble. You may find the equations ( derive them!) useful: $$\overline{\Delta E})^2\,=\,-\left(\frac{\partial\overline{E}}{\partial\beta}\right)_{\mu,V}\,+\,\frac{\mu}{\beta}\left(\frac{\partial \overline{E}}{\partial \mu}\right)_{T,V} $$ $$ \left(\frac{\partial \overline{E}}{\partial T}\right)_{\mu,V}\,=\,\left(\frac{\partial \overline{E}}{\partial T}\right)_{\overline{N},V}\,+\,\left(\frac{\partial \overline{E}}{\partial\overline{N}}\right)_{T,V}\left(\frac{\partial \overline{N}}{\partial T}\right)_{\mu,V} $$ $$ \left(\frac{\partial \mu}{\partial T}\right)_{\overline{N},V} \,=\,-\frac{\left(\frac{\partial \overline{N}}{\partial T}\right)_{\mu,V}}{\left(\frac{\partial \overline{N}}{\partial \mu}\right)_{T,V}} $$ The Maxwell's relation $$ \left(\frac{\partial\mu}{\partial T}\right)_{\overline{N},V}\,=\,-\left(\frac{\partial S}{\partial \mu}\right)_{T,V} $$ Also the relation derived in class ( you can assume this ) $$ \overline{\Delta N^2}\,=\,\left(\frac{\partial\overline {N}}{\partial\mu}\right)_{T,V} $$