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...

(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 $r=\infty .$