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The tension $\tau$ in an elastic rubber band is given by $$ \tau\,=\,aT\left(\frac{L}{L_0(T)}-\,\left(\frac{L_0(T)}{L}\right)^2\right),$$ where $a$ is a constant, $L_0(T)$is the unstretched length at zero tension, and is a function of temperature only.

• Write the first law using the work done when it is elongated and gets a supply of heat. ( Be careful of signs!)
• Use the first law to write $dF$, where $F$ is the free energy of the rubber band.
• Solve for the free energy $F$ and show that $$ F(T,L)\,-\,F(T,L_0(T))\,=\,aT\left(\frac{L^2}{2L_0(T)}\,+\,\frac{L_0(T)^2}{L^2}\,-\,\frac{3L_0(T)}{2}\right)$$ and the entropy $S$ $$ S(T,L)\,-\,S(T,L_0(T))\,=\,a\left(\frac{3L_0}{2}\,-\,\frac{L_0^2}{L}\,-\,\frac{L^2}{2L_0}\right)\,-\,aT\left(\frac{3}{2}\,-\,\frac{2L_0}{L}\,+\,\frac{L^2}{2L_0^2}\right)\frac{dL_0(T)}{dT} $$
• Find the heat $Q$ transferred to the elastic band when it is stretched from $L_0$ to $L$ isothermally.
• Show that $$ \left(\frac{\partial T}{\partial L}\right)_S \,=\,\frac{aTL_0^2}{c_LL^2}\left(-1\,+\,\left(\frac{L}{L_0}\right)^3\,+\,\frac{Ta}{L_0}\frac{dL_0}{dT}\left(2\,+\,\left(\frac{L}{L_0}\right)^3\right)\right)$$ where $$c_L\,=\,\left(\frac{DQ}{\partial T}\right)_L.$$

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The electrostatic energy  associated with continuous charge distribution is shown to correspond to  energy \(\frac{\epsilon_0}{2} |\vec E|^2\) per unit volume.

The electrostatic  energy of a continuous charge distribution is defined as the energy required to assemble the charges at  infinity into the positions as in the given distribution. For a continuous charge distribution it is shown to be \( \dfrac{\epsilon_0}{2}\iiint(\vec E\cdot\vec E) dV\) . Thus a volume of space having nonvanishing electric field has energy density  \(\dfrac{\epsilon_0}{2}(\vec E\cdot\vec E)\).The expression for the electrostatic energy reduces to the usual answer \(\frac{1}{2}  CV^2\) for a charged parallel plate capacitor. For a  uniformly charged sphere of radius \(R\) the electrostatic energy is proved to be equal to \(\frac{3}{5}\Big(\frac{Q^2}{4\pi\epsilon_0 R^2} \Big)\).   

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)

Classical Mechanics               July 3, 2019
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Test-I

• For a system the Hamiltonian is given to be \begin{equation*} H(q,p) = \frac{p^\beta}{\beta} + V(q) \end{equation*}
  1. Obtain Hamilton's equations of motion.\hfill[3]
  2. Show that the Lagrangian of the system is given by [4] \[L = \frac{q^\alpha}{\alpha} -V(q)\] where \(\alpha\) is given by \[ \frac{1}{\alpha} + \frac{1}{\beta}=1\]
• Consider a particle of mass $m$ moving in two dimensions in a potential The equation of motion for small oscillations are given to be \begin{eqnarray} m\ddot{x} = \frac{1}{2}\big( 3x + y \big)\\ m\ddot{y} = \frac{1}{2}\big(x + 3 y \big)\\ \end{eqnarray}                                                                                                                       [5+5+5]
  1. Find the normal frequencies of vibration in (b).
  2. Obtain expressions for \(x,y\) in terms of normal coordinates.
  3. Write the Lagrangian in terms of normal coordinates and verify that it takes the form \[{\Lsc} = \frac{1}{2}(\omega_1^2\dot{Q}_1^2+\omega_2^2\dot{Q}_2^2) - \frac{1}{2}(Q_1^2+Q_2^2)\]
• 1. Find a differential equation for \(f(P)\) so that the transformation[3+2+3] \begin{equation*} q=\frac{f(P)}{m\omega} \sin Q, \quad p= f(P) \cos Q \end{equation*} may be a canonical transformation.
  2. Solve the differential equation you get subject to condition \(f(0)=0\) and show that \[ f= \sqrt{2m\omega P}\]
  3. Obtain the type 1 generator for this transformation.