Chapter 6: Problem 9
Two equal charges \(q\) are located at the points \((\pm a, 0,0)\), and two charges \(-q\) at \((0, \pm a, 0)\). Find the leading term in the potential at large distances, and the corresponding electric field.
Chapter 6: Problem 9
Two equal charges \(q\) are located at the points \((\pm a, 0,0)\), and two charges \(-q\) at \((0, \pm a, 0)\). Find the leading term in the potential at large distances, and the corresponding electric field.
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Get started for freeThe potential \(\phi(\boldsymbol{r})=\left(q / 4 \pi \epsilon_{0} r\right) \mathrm{e}^{-\mu r}\) may be regarded as representing the effect of screening of a charge \(q\) at the origin by mobile charges in a plasma. Calculate the charge density \(\rho\) (at points where \(r \neq 0\) ) and find the total charge throughout space, excluding the origin.
Show that the moment of the Earth's gravitational force may be written in the form $$ m \boldsymbol{r} \wedge \boldsymbol{g}=\frac{3 G M m a^{2} J_{2}}{r^{5}}(\boldsymbol{k} \cdot \boldsymbol{r})(\boldsymbol{k} \wedge \boldsymbol{r}) $$ Consider a satellite in a circular orbit of radius \(r\) in a plane inclined to the equator at an angle \(\alpha\). By introducing a pair of axes in the plane of the orbit, as in \(\S 5.6\) (see Fig. 5.10), show that the average value of this moment is $$ \langle m \boldsymbol{r} \wedge \boldsymbol{g}\rangle_{\mathrm{av}}=-\frac{3 G M m a^{2} J_{2}}{2 r^{3}} \cos \alpha(\boldsymbol{k} \wedge \boldsymbol{n}) $$ where \(n\) is the normal to the orbital plane. Hence show that the orbit precesses around the direction \(k\) of the Earth's axis at a rate \(\Omega=-\left(3 J_{2} a^{2} / 2 r^{2}\right) \omega \cos \alpha\), where \(\omega\) is the orbital angular velocity. Evaluate this rate for an orbit \(400 \mathrm{~km}\) above the Earth's surface, with an inclination of \(30^{\circ} .\) Find also the precessional period.
Find the energy stored in a sphere of charge \(q\) and radius \(a\) with uniform charge density, and show that infinite energy is required to compress the sphere to a point. Find also the stored energy in the case where the charge is uniformly spread over the surface of the sphere.
*Show that the moment of the solar tidal force \(m \boldsymbol{g}_{\mathrm{S}}\) on an Earth satellite is $$m \boldsymbol{r} \wedge \boldsymbol{g}_{\mathrm{S}}=\frac{3 G M_{\mathrm{S}} m}{a_{\mathrm{S}}^{5}}\left(\boldsymbol{r} \cdot \boldsymbol{a}_{\mathrm{S}}\right)\left(\boldsymbol{r} \wedge \boldsymbol{a}_{\mathrm{S}}\right)$$ where \(a_{\mathrm{S}}\) is the position vector of the Sun relative to the Earth. Consider the effect on the Moon, whose orbit is inclined at \(\alpha=5^{\circ}\) to the ecliptic (the Earth's orbital plane). Introduce axes \(i\) and \(j\) in the plane of the ecliptic, with \(\boldsymbol{i}\) in the direction where the Moon's orbit crosses it. Write down the positions of the Sun and Moon, relative to Earth, as functions of time. By averaging both over a month and over a year (i.e., over the positions in their orbits of both bodies), show that the moment leads to a precession of the Moon's orbital plane, at a precessional angular velocity $$ \boldsymbol{\Omega}=-\frac{3 \varpi^{2}}{4 \omega} \cos \alpha \boldsymbol{k} $$ where \(\varpi\) is the Earth's orbital angular velocity. Compute the precessional period. (Note that this calculation is only approximately correct.)
Show that the work done in bringing two charges \(q_{1}\) and \(q_{2}\), initially far apart, to a separation \(r_{12}\) is \(q_{1} q_{2} / 4 \pi \epsilon_{0} r_{12}\). Write down the corresponding expression for a system of many charges. Show that the energy stored in the charge distribution is $$ V=\frac{1}{2} \sum_{j} q_{j} \phi_{j}\left(\boldsymbol{r}_{j}\right) $$ where \(\phi_{j}\left(\boldsymbol{r}_{j}\right)\) is the potential at \(\boldsymbol{r}_{j}\) due to all the other charges. Why does a factor of \(\frac{1}{2}\) appear here, but not in the corresponding expression for the potential energy in an external potential \(\phi(\boldsymbol{r}) ?\)
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