[2019EM/HMW-03]

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 \(R\).