Browse & Filter

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Problems with cylindrical symmetry can be solved by separating the variables of the Poisson equation in cylindrical coordinates.  The separation of variables for this class of problems and boundary conditions are explained.

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.

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.

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qm-lec-17003

Consider a paramagnetic system, with variables magnetization $M$, the magnetic field $B$ and absolute temperature $T$. ( We assume it's dependence on pressure as negligible). The equation of state is ( which will be obtained from statistical mechanics later in the course) is
$$ M\,=\,C\frac{B}{T}, $$
where $C$ is a constant ( referred to as the Curie constant, who had experimentally obtained this relation.
The system's internal energy is ( for a one-dimensional system)
$$ U\,=\,-MB.$$
The work done on the system by external surrounding is $-MdB$

(a) Write the expression for $DQ$ in terms of $dM$ and $dB$

(b) Write the equation for entropy change $dS$ in terms of $dM$ and $dB$

(c) Obtain the entropy $S$