diff --git a/_psets/6.md b/_psets/6.md index 611e574f3d70881c0adfc9d216df46466a34df9e..e2a2ada2c723ed2136e35832286e7106fce04df5 100644 --- a/_psets/6.md +++ b/_psets/6.md @@ -7,7 +7,109 @@ title: Problem Set 6 {:.question} Find the electric field for an infinitesimal dipole radiator. -So much math... check back later. +This problem is most naturally expressed in spherical coordinates. We know that the vector potential +is + +$$ +\begin{align*} +A_r &= \mu_0 \frac{I_0 d e^{-i k r}}{4 \pi r} \cos \theta \\ +A_\theta &= -\mu_0 \frac{I_0 d e^{-i k r}}{4 \pi r} \sin \theta \\ +A_\phi &= 0 +\end{align*} +$$ + +So the electric field is "simply" + +$$ +E = \frac{1}{i \omega \mu_0 \epsilon_0} \nabla ( \nabla \cdot A) - i \omega A +$$ + +First let's find $$\nabla \cdot A$$. + +$$ +\begin{align*} +\nabla \cdot A +&= \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 A_r) + + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta} ( A_\theta \sin \theta ) + + \frac{1}{r \sin \theta} \frac{\partial A_\phi}{\partial \phi} \\ +&= \frac{\mu_0 I_0 d}{4 \pi} \left( + \frac{\cos \theta}{r^2} \frac{\partial}{\partial r} \left( r e^{-ikr} \right) + - \frac{e^{-ikr}}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \sin^2 \theta \right) \\ +&= \frac{\mu_0 I_0 d}{4 \pi} \left( \frac{\cos \theta}{r^2} \left( e^{-ikr} - r i k e^{-ikr} \right) + - \frac{e^{-ikr}}{r^2 \sin \theta} \left( 2 \sin \theta \cos \theta \right) \right) \\ +&= \frac{\mu_0 I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( + \frac{1}{r^2} - \frac{ik}{r} - \frac{2}{r^2} \right) \\ +&= - \frac{\mu_0 I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( \frac{1}{r^2} + \frac{ik}{r} \right) +\end{align*} +$$ + +For any function $$f(r, \theta, \phi)$$ in spherical coordinates, + +$$ +\nabla f = \frac{\partial f}{\partial r} \hat{r} + + \frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\theta} + + \frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \hat{\phi} +$$ + +So the three components of $$\nabla (\nabla \cdot A)$$ are + +$$ +\begin{align*} +[\nabla (\nabla \cdot A)]_r +&= \frac{\partial (\nabla \cdot A)}{\partial r} \\ +&= - \frac{\mu_0 I_0 d}{4 \pi} \cos \theta \left( + \left( \frac{-2}{r^3} - \frac{ik}{r^2} \right) e^{-ikr} + + \left( \frac{1}{r^2} + \frac{ik}{r} \right) (-ik) e^{-ikr} \right) \\ +&= - \frac{\mu_0 I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( + \frac{-2}{r^3} - \frac{ik}{r^2} - \frac{ik}{r^2} + \frac{k^2}{r} \right) \\ +&= \frac{\mu_0 I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( + \frac{2}{r^3} + \frac{2ik}{r^2} - \frac{k^2}{r} \right) \\ + +[\nabla (\nabla \cdot A)]_\theta +&= \frac{1}{r} \frac{\partial (\nabla \cdot A)}{\partial \theta} \\ +&= - \frac{\mu_0 I_0 d}{4 \pi} e^{-ikr} \left( \frac{1}{r^3} + \frac{ik}{r^2} \right) + \frac{\partial}{\partial \theta} \cos \theta\\ +&= \frac{\mu_0 I_0 d}{4 \pi} e^{-ikr} \left( \frac{1}{r^3} + \frac{ik}{r^2} \right) \sin \theta \\ + +[\nabla (\nabla \cdot A)]_\phi &= 0 +\end{align*} +$$ + +Putting these together (and recalling that $$k = \omega \sqrt{\mu_0 \epsilon_0}$$) we find that + + +$$ +\begin{align*} +E_r +&= \frac{1}{i \omega \mu_0 \epsilon_0} \frac{\mu_0 I_0 d}{4 \pi} \cos \theta e^{-ikr} + \left( \frac{2}{r^3} + \frac{2ik}{r^2} - \frac{k^2}{r} \right) + - i \omega \frac{\mu_0 I_0 d}{4 \pi} \frac{\cos \theta}{r} e^{-ikr} \\ +&= \frac{\mu_0 I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( \frac{1}{i \omega \mu_0 \epsilon_0} + \left( \frac{2}{r^3} + \frac{2ik}{r^2} - \frac{k^2}{r} \right) - \frac{i \omega}{r} \right) \\ +&= \frac{I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( \frac{2}{i \omega \epsilon_0 r^3} + + \frac{2k}{\omega \epsilon_0 r^2} - \frac{k^2}{i \omega \epsilon_0 r} + - \frac{i \omega \mu_o}{r} \right) \\ +&= \frac{I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( \frac{2}{i \omega \epsilon_0 r^3} + + \frac{2 \omega \sqrt{\epsilon_0 \mu_0}}{\omega \epsilon_0 r^2} + - \frac{\omega^2 \epsilon_0 \mu_0}{i \omega \epsilon_0 r} - \frac{i \omega \mu_o}{r} \right) \\ +&= \frac{I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( \frac{2}{i \omega \epsilon_0 r^3} + + \frac{2}{r^2} \sqrt{\frac{\mu_0}{\epsilon_0}} + + \frac{i \omega \mu_0}{r} - \frac{i \omega \mu_o}{r} \right) \\ +&= \frac{I_0 d}{4 \pi} \cos \theta e^{-ikr} \left( \frac{2}{i \omega \epsilon_0 r^3} + + \frac{2}{r^2} \sqrt{\frac{\mu_0}{\epsilon_0}} \right) \\ + +E_\theta +&= \frac{1}{i \omega \mu_0 \epsilon_0} \frac{\mu_0 I_0 d}{4 \pi} \sin \theta e^{-ikr} + \left( \frac{1}{r^3} + \frac{ik}{r^2} \right) + + i \omega \frac{\mu_0 I_0 d}{4 \pi} \frac{\sin \theta}{r} e^{-ikr} \\ +&= \frac{I_0 d}{4 \pi} \sin \theta e^{-ikr} \left( \frac{1}{i \omega \epsilon_0 r^3} + + \frac{k}{\omega \epsilon_0 r^2} + \frac{i \omega \mu_0}{r} \right) \\ +&= \frac{I_0 d}{4 \pi} \sin \theta e^{-ikr} \left( \frac{1}{i \omega \epsilon_0 r^3} + + \frac{1}{r^2} \sqrt{\frac{\mu_0}{\epsilon_0}} + \frac{i \omega \mu_0}{r} \right) \\ + +E_\phi &= 0 +\end{align*} +$$ ## (8.2)