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Electrodynamics                                                                          Oct 17, 2018
                                                Tutorial-V (VI)

• A cylindrical resistor of length \(\ell\), radius \(a\) and resistivity $\varrho$ carrying a current \(i\).
  1. Show that the Poynting vector $\vec{S}$ at the surface of the resistor is everywhere directed to the normal to the surface, as shown,
  2. Show the rate $\rho$ at which energy flows into the resistor through its cylindrical surface, calculated by integrating the Poynting vector over this surface, is equal to the rate at which Joule heat is produced, i.e. \begin{equation*} \int\vec{S}\cdot\vec{dA}=i^2R \end{equation*}
• Figure 1 shows a parallel-plate capacitor being charged,
  1. Show that the Poynting vector \(S\) points everywhere radially into the cylindrical volume.
  2. Show that the rate \(P\) at which energy flows into this volume, calculated by integrating the Poynting vector over the cylindrical boundary of this volume, is equal to the rate at which the stored electrostatic energy increases; that is, that \[\int \vec{S}\cdot\vec{dA} = A d \frac{1}{2}\frac{d}{dt}(\epsilon \vec{E}^2)\],where \(Ad\) is the volume of the capacitor and \(\frac{1}{2} \epsilon_0\vec{E}^2\) is the energy density for all points within that volume. This analysis shows that, according to the Poynting vector point of view, the energy stored in a capacitor does not enter it through the wires but through the space around the wires and the plates. \{Hint: To find \(S\) we must first find \(B\), which is the magnetic field setup by the displacement current during the charging process; Ignore fringing of the lines of \(E\).)\}
  
• A coaxial cable (inner radius \(a\) and outer radius \(b\)) is used as a transmission line between a battery $\epsilon$ and a resistor \(R\),

1. [(a)] Calculate $\vec{E}, \vec{B}$ for $a < r < b$
2. [(b)] Calculate Poynting vector S for $a< r < b.$
3. [(c)] By suitably integrating the Poynting vector, Show that the total power flowing across the annular cross section $a<r<b$ is="" $\epsilon^{2}="" r$.="" this="" reasonable="" ?="" <="" li="">
4. [(d)] Show that the direction of $\vec{S}$ is always away from the battery to resistor, no matter which way is the battery connected.

Elementary Particle Physics

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qm-lec-18004
Some algebraic manipulations
Large r expansion of the formal solution

Consider N particles having volume V with the energy per particle be given by the relativistic expression $\sqrt{c^2\,p^2\,+\,m^2\, c^4}$. It is in equilibrium with a reservoir at temperature T. Assuming $\frac{p}{mc}$ is small, find the leading correction to the (a) Canonical partition function and (b) Entropy in terms of V,N and T up to an overall constant.

Classical Mechanics               June 17, 2019   

Quiz-II

1. For a system with coordinates \(q_k\) and canonical momenta \(p_k\), compute the Poisson brackets \begin{equation} \{q_k, F(q,p)\}_\text{PB}; \qquad \qquad\{p_k, G(q,p)\}_\text{PB} \end{equation}
2. Set up Lagrangian for a coupled pendulum shown in figure below. Assume that the system while oscillating remains in a vertical plane.

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qm-lec-16002

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We discuss how Maxwell's addition of a displacement current in the fourth equation.