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The scattering by the dielectric sphere of Problem $10.4$ was treated as purely electric dipole scattering. This is adequate unless it happens that the real dielect...

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The scattering by the dielectric sphere of Problem $10.4$ was treated as purely electric dipole scattering. This is adequate unless it happens that the real dielectric constant $e / \epsilon_{0}$ is very large. In these circumstances a magnetic dipole contribution. even though higher order in $k R$, may be important. (a) Show that the changing magnetic flux of the incident wave induces an az. muthal current flow in the sphere and produces a magnetic dipole moment, $$ \mathrm{m}=\frac{i 4 \pi \si

The scattering by the dielectric sphere of Problem $10.4$ was treated as purely electric dipole scattering. This is adequate unless it happens that the real dielectric constant $e / \epsilon_{0}$ is very large. In these circumstances a magnetic dipole contribution. even though higher order in $k R$, may be important. (a) Show that the changing magnetic flux of the incident wave induces an az. muthal current flow in the sphere and produces a magnetic dipole moment, $$ \mathrm{m}=\frac{i 4 \pi \sigma Z_{0}}{k \mu_{0}}(k R)^{2} \frac{R^{3}}{30} \mathbf{B}_{\text {inc }} $$ (b) Show that application of the optical theorem to the coherent sum of the electric and magnetic dipole contributions leads to a total cross section, $$ \sigma_{i}=12 \pi R^{2}\left(R Z_{0} \sigma\right)\left[\frac{1}{\left(\epsilon_{r}+2\right)^{2}+\left(Z_{0} \sigma / k\right)^{2}}+\frac{1}{90}(k R)^{2}\right] $$



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(III) The dipole moment, considered as a vector, points from the negative to the positive charge. The water molecule, Fig. $32,$ has a dipole moment $\vec{\mathbf{p}}$ which can be considered as the
vector sum of the two dipole moments $\vec{\mathbf{p}}_{1}$ and $\vec{\mathbf{p}}_{2}$ as shown. The distance between each $\mathrm{H}$ and the $\mathrm{O}$ is about $0.96 \times 10^{-10} \mathrm{m} ;$ the lines joining the center of the $\mathrm{O}$ atom with each $\mathrm{H}$ atom make an angle of $104^{\circ}$ as shown, and the net dipole moment has been measured to be $p=6.1 \times 10^{-30} \mathrm{C} \cdot \mathrm{m} .(a)$ Determine the effective charge $q$ on each $\mathrm{H}$ atom. $(b)$ Determine the electric potential, far from the molecule, due to each dipole, $\vec{\mathbf{p}}_{1}$ and $\vec{\mathbf{p}}_{2},$ and show that $V=\frac{1}{4 \pi \epsilon_{0}} \frac{p \cos \theta}{r^{2}}$ where $p$ is the magnitude
of the net dipole moment, $\ddot{\mathbf{p}}=\vec{\mathbf{p}}_{1}+\vec{\mathbf{p}}_{2},$ and $V$ is the total potential due to both $\vec{\mathbf{p}}_{1}$ and $\vec{\mathbf{p}}_{2} .$ Take $V=0$ at $r=\infty .$


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