[2018EM/Final-B-Sol]

Electrodynamics Nov                                                                                28, 2018
Final Examination :: Part-B

[\(\oslash\)] Attempt any ten questions. If you attempt all questions, the best ten answers will be evaluated.
[\(\oslash\)] Each question carries three marks.
[\(\oslash\)] The total marks awarded will be scaled to fifteen for final evaluation.
[\(\oslash\)] Full marks will awarded only if answers are correct, clear, precise and complete.

• For which of the following modifications, if any, in the form of Coulomb force will Gauss law remain valid?
  1. The magnitude of the force varies as \(r^{2-\epsilon}\), \(\epsilon >0\).
  2. The direction of the force is not along the line joining the two charges.
  3. The magnitude of the force depends on the direction of the line joining the point charges.
  For each of the above statements write YES/NO as your answer.
• Which statements that are true in electrostatics remain valid in electrodynamics? Write as many statements as you can.
• Which are the statements that are true in electrostatics but not true in electrodynamics? Write statements as many as you can.
• State the uniqueness theorems for solutions of Laplace equation. Be as precise as you can.
• Find the net force on a point charge \( q\) placed symmetrically at a distance \( d\) from two perpendicular grounded conducting infinite planes. If you wish, you may take the planes to be \(X-Z\) and \(Y-Z\) planes.
• Write the electric potential at a point $\vec{r}$ for a system of three charges $q$, $-3q$ and $2q$ placed at positions $x=-3a,a,3a$ on the \(X-\) axis, respectively. Assume a form $$ \phi(r) \approx \frac{A}{r} + \frac{B}{r^2}+ \frac{C}{r^3} + O(r^{-4}) $$ for the potential at large distance. What can you say about the numerical values of constants \(A,B,C\) {\it without expanding the potential expression for large} \(r\)? Is your answer independent of translations of the coordinate axes? Give a brief explanation for your answers.
• Find the magnetic field at a point \(\vec{r}\) due to three infnite wires, along the coordinate axes, each carrying a current \(I\). Do not give the final result for the components of \(B\) in terms of \((x,y,z)\). Express the final answer for \(\vec{B}\) in vector notation in terms of \(\vec{r}\), and unit vectors \(\hat{i},\hat{j}, \hat{k}\) along the coordinate axes.
• A light cylindrical shell of aluminium is lying on a smooth horizontal table, take it to be \(X-Y\) plane). The axis of the cylinder lies along the \(X-\)axis. A bar magnet is held above the shell at some distance.\\ {\tt What happens when the bar magent is moved in the direction of positive \(Y\) axis.} Write a brief explanation of your answer.
• Write the transformation property for the potentials \(\phi, \vec{A}\) under from a frame \(K\) to frame \(K'\). It is given that frame \(K'\) moves along the \(X\) axis with velocity \(v\). Starting from the electrostatic potential due to a point charge at rest in frame \(K'\), find the potentials \(\phi(\vec{r},t), \vec{A}(\vec{r},t)\) in the frame \(K\).
• Give an example of a system of point charges which has zero dipole moment and nonvanishing quadrupole moment. Construct the example in such a way that the quadrupole moment changes under a translation of the coordinate axes.
• Two spheres of conducting material have radii 1 cm and 10 cm and carry charge $100$\,C and $1$\,C, respectively. The separation between the centers of the two spheres is 10 m.
  1. What is the potential of each sphere.
  2. Find the charges on the two spheres, if the two spheres are connected by a fine wire.
  3. Are the values obtained by you exact or approximate(give reasons)?

• Solution:- The potential of each sphere will be constant and we assume that the potential on the surface of the each sphere can be approximately found by taking the charged spheres as point charges. An exact solution cannot be given by any of the methods covered in the class. Let $\vec{r_1}, \vec{r_2}$ denote the position vectors of the centers of the two spheres. We shall assume that the potential on the surface of each of the two spheres can be computed by assuming second seond sphere to be a point charge. Also the total potential is a superposition of potentials in this approximation.
  1. [(a)] Before the two spheres are connected the potential of the first sphere is $$\phi_1 \approx \frac{1}{4\pi\varepsilon_0}\left( \frac{q_1}{r_1} +\frac{q_2}{|\vec{r_1}-\vec{r}_2|}\right)$$ and the potential of the second sphere is $$\phi_2\approx\frac{1}{4\pi\varepsilon_0}\left( \frac{q_1}{|\vec{r}_1-\vec{r}_2|} +\frac{q_2}{r_2}\right) $$ Taking the values $r_1=1/10,\,r_2=1/100, |\vec{r}-1-\vec{r}_2|=10$m and initial values of the charges are $q_1=100, ~~~q_2=1,$ $$ \therefore \phi_1=(9\times10^9)\left(\frac{100}{1/100} + \frac{1}{10}\right) \approx 9\times10^{13} V $$ $$\phi_2 = (9\times10^9)\left(\frac{100}{10} + \frac{1}{1/10}\right) \approx 18\times 10^{10} \, \mbox{\rm V} $$
  2. [(b)] When the two spheres are connected, the charge will flow from one to the other sphere till their potentials become equal. So if $q_1$ and $q_2$ are the charges after connecting the two spheres, $\phi_1=\phi_2$ and we get $$ \frac{q_1}{r_1} + \frac{q_2}{d}= \frac{q_1}{d}+\frac{q_2}{r_2} $$ Also we have the total charge $$ q_1+q_2=101 \times 10^{-8}$$ Thus we get \begin{eqnarray} q_1 &\approx&\frac{101}{122}\times 11 \approx 9.107 \mbox{ coul}\\ q_2&\approx& \frac{101}{122}\times 111 \approx 91.893 \mbox{ coul} \end{eqnarray}