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Electrodynamics                                                                     March 8, 2019                                                                                  Tutorial-III

• The potential takes the constant value \(\phi_0\) on the closed surface \(S\) which bounds the volume \(V\). The total charge inside V is \(Q\). There is no charge anywhere else. Show that the electrostatic energy contained in the space outside of S is \(\displaystyle U_\text{E}(out)=\frac{1}{2} Q\phi_0\)
• The line segment between \(A\) to \(B\) in the diagram below carries a uniform charge per unit length \(\lambda\). The vector \(\vec{a}\) is coincident with the segment. The vectors \(\vec{b}\) and \(\vec{c}\) point from the observation point \(\vec{r}\) to the beginning and end of \(\vec{a}\), respectively.Evaluate the integral for the potential in a coordinate-free manner by parameterizing the line source using a variable \(s\) which is zero at the point on the charged segment that is closest to \(\vec{r}\). Show thereby that \begin{eqnarray} \phi(\vec{r}) &=& \frac{\lambda}{4\pi\epsilon_0}\ln {\left|\frac{\vec{b}\cdot\vec{a}}{a} +\sqrt{\left(\frac{\vec{b}\cdot\vec{a}}{a}\right)^2 + \frac{|\vec{b}\times\vec{a}|^2}{a^2} } \right| }\nonumber\\ && -\frac{\lambda}{4\pi\epsilon_0} \ln {\left|\frac{\vec{c}\cdot\vec{a}}{a} +\sqrt{\left(\frac{\vec{c}\cdot\vec{a}}{a}\right)^2 + \frac{|\vec{b}\times\vec{a}|^2}{a^2} } \right| } \end{eqnarray}
• Suppose the electrostatic potential of a point charge were \[V (r) =\frac{q}{4\pi\epsilon_0}\frac{1}{r^{(1+\epsilon)}}\] rather than the usual Coulomb formula.
  • Find the potential \(V(r)\) at a point at a distance \(r\) from the center of a spherical shell of radius \(R > r\) with uniform surface charge per unit area \(\sigma\). Check the Coulomb limit \(\epsilon = 0\).
  • To first order in \(\epsilon\), show that \begin{equation} \frac{V(R)-V(r)}{V(R)} =\frac{\epsilon}{2}\left[\frac{R}{r}\ln\frac{R+r}{R-r} - \ln \frac{4R^2}{R^2-r^2}\right] \end{equation}
  Since the time of Cavendish, formulas like this one have been used in experimental tests of the correctness of Coulomb’s law.
• Find the primitive, Cartesian monopole, dipole, and quadrupole moments for each of the following charge distributions. Use the geometrical center of each as the origin.
  • Two charges \(+q\) at two diagonal corners of a square \((\pm a, \pm a, 0)\) and two minus charges \(−q\) at the two other diagonals of the square \((\pm a, \mp a, 0)\).
  • A line segment with uniform charge per unit length λ which occupies the interval \(-\ell \le z \le \ell\).` .
  • An origin-centered ring in the \(X-Y\) plane with uniform charge per unit length \(\lambda\) and radius \(R\).

Electrodynamics                                                Apr 8, 2019
                                        Quiz-X

• At the upper surface of the Earth’s atmosphere, the time-averaged magnitude of the Poynting vector \(<S> =1.35^10^{W/m}^2\) is referred to as

  the solar constant.

  1. Assuming that the Sun’s electromagnetic radiation is a plane sinusoidal wave, what are the magnitudes of the electric and magnetic fields?
  2. What is the total time-averaged power radiated by the Sun? The mean Sun-Earth distance is \(R =1.50\times10^{11}\) m .

• Following equation contains the complete information about the electromagnetic wave \begin{equation*} \vec{E}(z,t)=E_0\sin(kz-\omega t)\hat{i} \end{equation*} Find the directions of wave propagation, wavelength, frequency, speed of propagation, magnetic field, the Potynting vector and the intensity of the wave.

Electrodynamics                                                Apr 19, 2019
                                     Quiz-VIII

Question 1:
A student was asked to find the electric field inside
a uniformly polarized sphere. The following answer was received.

Sample Answer:


The answer to the above question as given in Griffiths is

\(\vec{E}=-\frac{\vec{P}}{3\epsilon_0}\)

1. Give your comments why is this answer wrong?
2. Give a counter example showing that the conclusion arrived by the student is incorrect.
3. Give steps to derive the correct answer

Question 2:
Consider a parallel plate capacitor with plates having charges \(Q\) and \(-Q\). An insulating medium fills the region between the plates. Assuming that the linear dimensions of the plates are much larger than the separation between the plates, draw a diagram to show

1. the distribution of free charges on the plates.
2. distribution of the bound charges on the surface of the dielectric material.
3. Use Gauss law to obtain \(\vec{D}\). Use your result for \(\vec{D}\) and obtain \(\vec{E}\), the potential difference between the plates and the capacitance.

 Electrodynamics                                              Apr 1, 2019

                                                 Quiz-VI

• Two infinite long wires, carrying currents $I$ and $2I$, pass through points $A,B$ separated by a distance $10L$, as shown in figure below.
  • Sketch a figure to show the directions of the magnetic field due the two wires at a point $P$ at distance $12L$ from the line $AB$.
  • Find the magnitude and direction of the resultant magnetic field due to the two wires at point $P$.
  
• A long straight wire carrying a current $2I$ is placed in the plane having a rectangular loop carrying current $3I$ as shown in the figure.
  • What are the magnitude and directions of magnetic force on the loop the two sides $AD$ and $BC$ due to the current in the straight wire?
  • Divide the side $AB$ into small line elements, write the force on a small line element and hence find the net magnetic force on the side $AB$ due the straight wire.
  • Compute the net force on the loop when $I=10$A, $L=10$ cm, $a=1$cm, $b=80$cm.
  

Electrodynamics                                                               Feb 1,2008

                                           2018 Quiz-II

1. Find the direction and magnitude of $\vec{E}$ at the center of a rhombus, with interior angles of $\pi/3$ and $2\pi/3$, with charges at the corners as shown in figure below. Assume that $ q= 1\times 10^{-8}$C, $a=5$cm 

2. Two spheres each of radius $R$ are placed so that they partially overlap. Thecharge densities in the overlap region is zero and in the two non overlappingregions is $+\rho$ and $-\rho$ respectively as shown in figure.The separation between the centres of the spheres is $D$.Show that the electric field in the overlap region is constant.

 Electrodynamics                                               March 11, 2019

                                      Quiz-IV

In case of a charge distribution having symmetry about \(z\)-axis,

\(Q_{xy}=Q_{yz}=Q_{zx}=0\) and \(Q_{yy}=Q_{xx}\) Also trace
\(Q_{xx}+Q_{yy}+Q_{zz}\) is always zero. Thus the quadrupole moment tensor can
be specified by a single number
\(Q= 2(Q_{zz}-Q_{xx})\). Calculate \(Q\) for six equal charges placed on corners
of a regular hexagon of sides \(a\).

The definition of quadrupole moment tensor components is
\begin{eqnarray}
Q_{xx} &=&\frac{1}{2} \sum_\alpha Q_\alpha (3x^2_\alpha - r^2_\alpha)\\
Q_{yy} &=& \frac{1}{2} \sum_\alpha Q_\alpha (3y^2_\alpha - r^2_\alpha)\\
Q_{zz}&=& \frac{1}{2} \sum_\alpha Q_\alpha (3z^2_\alpha - r^2_\alpha)
\end{eqnarray}
In this case \(Q_{xz}=Q_{yz}=0\) because all charges are in \(XY\) plane
\((z=0)\). Also \(Q_{xy}=0\) due to symmetry reflection symmetry in the \(X,
Y\) axes.
For all the charges \(z=0, r=a\). Hence
\begin{equation*}
\sum_{\alpha=1}^6 z^2=0 \qquad \sum_{\alpha=1}^6 r^2= 6a^2
\end{equation*}
For four charges \(x^2= \frac{3a^2}{4}\) and for two charges \(x=0\). Therefore
\begin{equation*}
\sum_{\alpha=1}^6 3 x^2 = 9a^2
\end{equation*}
Thus we get
\begin{eqnarray}
Q_{xx} &=& \frac{1}{2} \sum_\alpha Q_\alpha (3x^2_\alpha - r^2_\alpha)=
Q (9a^2 - 6a^2) = 3Qa^2 \\
Q_{zz} &=& \frac{1}{2} \sum_\alpha Q_\alpha (3z^2_\alpha - r^2_\alpha)= - 6Qa^2
\end{eqnarray}
and the final answer is
\[Q= 2(-6Qa^2- 3Qa^2) = -18 Qa^2.\]

Electrodynamics                                                  April 3, 2022
                                            Quiz-I

 Two grounded infinite conducting planes are kept along the XZ and Y Z planes, see Fig.2. A charge q is placed at (4,3) find the force acting on the charge q.

                                              Classical Mechanics               May 22, 2019   

Quiz-I

A a pendulum having a bob of mass \(m\) is attached to a mass \(M\) by a string of length \(\ell\) passing through a linear slit parallel to \(X\) axis. The mass \(M\) is kept on a frictionless table and is free to move along the \(X\) axis.

• Choose \(x\) and \(\theta\) as generalized coordinates and write Lagrangian for the two systems in figures below.
• For each system state if it will have a zero frequency of oscillations or not. Give a short explanation for your explanation.