[2019EM/Final]

TIME :3hrs                   ELECTRODYNAMICS                                MM:100                                         

                               End Semester Examination

                                       ATTEMPT ANY FIVE QUESTIONS.

1. Find the potential due to a sphere carrying a uniform polarization \(\vec{P}\). The centre of the sphere is at the origin and radius of the sphere is \(R\). What is the electric field at a point (i) inside the sphere (ii) outside the sphere?
2. 1. Give an example of a charge configuration such that its dipole moment is zero and quadrupole moment depends on the choice of origin.
   2. An alpha particle travels in a circular path of radius $0.45$m in a magnetic field with $B=1.2$w/m$^2$. Calculate  (i) its speed (ii) its period of revolution, and (iii) its kinetic energy.Mass of alpha particle = \(6.64424. 10^{-27}\)kg \(\approx 4\times M_p= 4\times938.27\) MeV.
   3. A circular coil is formed from a wire of length $L$ with $n$ turns. The coil carries a current $I$ and is placed in an external uniform magnetic field $B$. Show that maximum torque developed is $\displaystyle\frac{IBL^2}{4n\pi}$.
   4. Give examples of at least six results/concepts that require modifications in time varying situation.                                                                            [4+5+5+6]
3. 1. Derive an expression for electrostatic energy of a charge distribution and hence show that electric field carries energy density \(\frac{\epsilon_0}{2}|\vec{E}|^2\).
   2. A conducting spherical shell carries a charge \(Q\), compute its electrostatic energy and hence obtain an expression for the capacitance of the shell.\hfill[10+10]
4. A rod of mass $m$ and length $\ell$ and resistance $R$ starts from rest and slides on two parallel rails of zero resistance as shown in figure 1. A uniform magnetic field fill the area and is perpendicular and out of the plane of the paper. A battery of of voltage $V$ is connected as shown in the figure 1.
   1. Argue that the net EMF in the loop is $V = Bv\ell$ when the rod has speed $v$.
   2. Write down $F = m\big(\dfrac{dv}{dt}\big)$ and integrate it so show that \begin{equation*}\label{EQ01} v(t) =\frac{V}{B\ell}\Big(1- \exp\Big(- \frac{B^2\ell^2 t}{mR}\Big)\Big). \end{equation*} Hint: Find the limiting speed and separate that out from the total $v$.
   3. What happens when the direction of magnetic field is reversed?            [8+8+4] 
5. 1. Show that, in absence of charges and currents, the electric and magnetic fields obey wave equation.
   2. State and prove important properties of plane wave solutions.
   3. Obtain an expression for energy density and intensity of plane waves.      [6+8+6]
6. An infinite rectangular hollow pipe is bounded by the planes \(x=\pm a,y=0, y=b\). The pipe extends to infinity in positive as well as negative \(Z\)- directions. The sides \(y=0, x=\pm a\) are grounded and the the side \(y=b\) is held at constant potential \(\phi_0\). Show that the potential inside the pipe is  \begin{equation*} \phi(x,y) = \phi_0\Big\{\frac{y}{b} +\frac{2}{\pi} \sum_{n=1}^\infty\frac{(-1)^n}{n} \frac{\cosh (n\pi x/b)}{\cosh(n\pi a/b)} \sin (n\pi y /b) \Big\} \end{equation*}