[2018EM/Final-A]

$ \newcommand {\pieps}{4 \pi \epsilon_0} $

Electrodynamics Nov 22, 2018
Part-A:: Take Home Final Examination 

• 1. Two long coaxial cylindrical shells of radii \(a\) and \(b\) are kept at the potentials \(V_a\) and \(V_b\), respectively. Using separation of variables solve the Laplace equation and find the potential at any point between the shells. [10]
  2. Laplace equation for potential \[\nabla^2 \phi =\rho_0(\vec{r})\] is to be solved outside a long cylinder. The radius of the cylinder is \(a\) and the axis of the cylinder lies along the \(z\)- axis. The potential is required to satisfy the boundary conditions\footnote{needs to be changed to \begin{equation*} \qquad \phi(\vec{r})\big|_{\rho\to\infty}=0. \end{equation*} \begin{equation*} \phi(\vec{r})\big|_{\rho=a}=0; \end{equation*} where \(\rho=\sqrt{x^2+y^2}\). Use a method, other than the separation of variables, to find the Green function for this problem. [10]
• Three point charges \((q,-2q,q)\) are located in a straight line. The separation of each charge \(q\) from the charge \(-2q\) is \(a\). The middle charge \((\-2q)\) is kept at the center of a conducting spherical shell of radius \(b\). Find the Green function for this problem. Write the charge density for the problem and use the Green function to solve the problem for the potential everywhere inside the sphere. [10+10]
• A massive spin one boson is like photon with mass. It is described by a four vector field \(V_\mu\). For a set of three massive vector bosons, collectively denoted as \(\vec{V}_\mu= \{V_{1\mu}, V_{2\mu}, V_{3\mu}\}\), the Lagrangian is given by \begin{equation*} {\mathcal L} =-\frac{1}{4}\vec{F}_{\mu\nu}\cdot\vec{F}^{\mu\nu} + \frac{1}{2} M^2\vec{V}_\mu \cdot \vec{V}_\mu. \end{equation*} where \begin{equation*} \vec{F}_{\mu\nu} = \partial_\mu \vec{V}_\nu - \partial_\nu \vec{V}_\mu + g \vec{V}_\mu \times \vec{V}_\nu. \end{equation*} Here operations dot \(\cdot\), and cross \(\times \), denote usual scalar and cross products of three vectors.
  1. Derive equation of motion for \(V_{j\alpha}\) and show that the equations of motion can be written as\null [15] \begin{equation*} \partial_\nu \vec{F}^{\nu\alpha}+ g\big(\vec{F}^{\nu\alpha}\times\vec{V}_\alpha\big) + M^2V^\alpha =0. \end{equation*}
  2. Verify that the Lagrangian invariant under rotations acting on the index \(k\), {\it i.e.} under continuous transformation \begin{eqnarray} \vec{V}'_\mu(x) &=& \vec{V}_\mu(x) +\delta \vec{V}_\mu(x),\nonumber\\ \delta \vec{V}_\mu(x) &=& \vec{\epsilon} \times \vec{V}_\mu (x) \nonumber. \end{eqnarray} where \(\vec{\epsilon}=(\epsilon_1, \epsilon_2, \epsilon_3)\) are infinitesimal parameters. Determine the corresponding conserved currents \(\vec{J}_{\mu}\) as given by the Noether's theorem and satisfying\null [15] \[ \partial\,^\mu \vec{J}_{\mu} = 0.\]
• A dipole \(\vec{d}\), rotates in a plane with constant angular velocity \(\Omega\). Choosing the plane of rotation as the \(x\)-\(y\) plane, find \null [20]
  1. the total power radiated.
  2. the angular distribution of power radiated averaged over one period of rotation.