[2018EM/HMW-03]

Electrodynamics                                                                                Sep 3, 2018

Tutorial III 

• 1. Determine the polarizability of two charges that are bound together by a harmonic potential with force constant k.
  2. How much work is done on water molecule if \(E\) is increased from zero to 100 volts/cm and the permanent dipoe is rotated from \(\theta=0\) to \(\theta=\pi\)?
• 1. Farad is a large unit of capacitance. Estimate the size of sphere having capacity 1 farad and compare your answer with the radius of Earth.
  2. If an electron is placed on a sphere with capacity \(C=1\), what is the energy required? What will be the energy required for a nano sized sphere?
• Two isolated conducting spheres of radii \(R_1\) and \(R_2\) are placed a distance \(L\) apart, with \(L> R_1\) and \(L > R_2\). A charge \(Q\) is placed on sphere 1. This configuration can be solved by successive approximations involving images, images of images, etc.
  1. To the lowest order in \(R_{1,2}/L\) calculate contribution to the force between the spheres. Carry the calculation one step farther,sufficient to identify the power of \(R_{1\text{or}2}/L\) of the next correction.
  2. If you calculated the force by setting it equal to the force between the image charges, your calculation was correct. Can you justify this statement? Although the answer "no" may be correct (i.e., you cannot justify this statement), it is not acceptable.
• The potentials \(\phi_1, \phi_2, ...\) of several conductors is a linear function of charges, \(Q_1, Q_2,...,\) on the conductors. Thus \[ p_j= \sum_k p_{jk}Q_k \] Find the electrostatic energy of concentric spheres shown in figure below, having charges \(Q_1, Q_2\) by finding the matrix \(\begin{pmatrix} p_{11}, p_{12} \\ p_{21},p_{22}\end{pmatrix}\). Answer:\( \frac{1}{8\pi\epsilon_0}\Big[\Big(\frac{1}{a}-\frac{1}{b} + \frac{1}{c}\Big) Q_1^2 + \frac{2Q_1Q_2}{c} + \frac{Q_2^2}{c} \Big]\)