diff --git a/_psets/6.md b/_psets/6.md new file mode 100644 index 0000000000000000000000000000000000000000..1574523c8e17536a8e7688ad36ec368e8bed93a3 --- /dev/null +++ b/_psets/6.md @@ -0,0 +1,259 @@ +--- +title: Problem Set 6 +--- + +## (8.1) + +{:.question} +Find the electric field for an infinitesimal dipole radiator. + +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) + +{:.question} +What is the magnitude of the Poynting vector at a distance of 1 km from an antenna radiating 1 kW of +power, assuming that it is an isotropic radiator with a wavelength much less than 1 km? What is the +peak electric field strength at that distance? + +Since it's an isotropic radiator, the Poynting vector must point along $$\hat{r}$$. Similarly its +magnitude must depend only on $$r$$, so we can compute its time average value by dividing the total +power by the total area. + +$$ +\begin{align*} +\langle |P| \rangle &= \frac{10^3 \si{W}}{4 \pi \cdot 10^6 \si{m^2}} \\ +&= \num{8e-5} \si{W/m^2} +\end{align*} +$$ + +Since $$\lambda << 1\si{km}$$, locally the radiation will look like a plane wave. Thus E and H are +perpendicular, and $$|H| = \sqrt{\epsilon_0/\mu_0} |E|$$. + +$$ +\begin{align*} +\langle |P| \rangle &= \langle |E| |H| \rangle \\ +&= \sqrt{\frac{\epsilon_0}{\mu_0}} \langle |E|^2 \rangle \\ +&= \frac{1}{2} \sqrt{\frac{\epsilon_0}{\mu_0}} E^2_\text{max} +\end{align*} +$$ + +Solving for $$E_\text{max}$$, + +$$ +\begin{align*} +E_\text{max} &= \left( 2 \sqrt{\frac{\mu_0}{\epsilon_0}} \langle P \rangle \right)^\frac{1}{2} \\ +&= \sqrt{2 \cdot 377 \si{\ohm} \cdot \num{8e-5} \si{W/m^2}} \\ +&= 0.24 \si{V/m} +\end{align*} +$$ + + +## (8.3) + +{:.question} +For what value of $$R_\text{load}$$ is the maximum power delivered to the load in Figure 8.3? + +By Ohm's Law $$V = I (R_r + R_l)$$ (where $$R_r$$ is the radiation resistance, and $$R_l$$ the load +resistance). Thus $$I = V / (R_r + R_l)$$ and the power dissipated in the load resistance is + +$$ +\begin{align*} +P &= I^2 R \\ +&= \frac{V^2 R_l}{(R_r + R_l)^2} +\end{align*} +$$ + +To maximize this we take the derivative + +$$ +\frac{d P}{d R_l} = V^2 ( (R_r + R_l)^{-2} - 2 R_l (R_r + R_l)^{-3} ) +$$ + +Setting this to zero implies $$R_r + R_l = 2 R_l$$, or + +$$ +R_l = R_r +$$ + + +## (8.4) + +{:.question} +For an infinitesimal dipole antenna, what are the gain and the area, and what is their ratio? + +From the text we know that + +$$ +\langle |P| \rangle += \frac{I_0^2 k^2 d^2}{32 \pi^2 r^2} \sqrt{\frac{\mu_0}{\epsilon_0}} \sin^2 \theta +$$ + +and + +$$ +W = \frac{I_0^2 \pi}{3} \sqrt{\frac{\mu_0}{\epsilon_0}} \left( \frac{d}{\lambda} \right)^2 +$$ + +Thus + +$$ +\begin{align*} +G &= \max_{\theta, \phi} \frac{\langle |P(r = 1, \theta, \phi)| \rangle}{W/4 \pi} \\ +&= 4 \pi \frac{I_0^2 k^2 d^2}{32 \pi^2} \sqrt{\frac{\mu_0}{\epsilon_0}} +\frac{3}{I_0^2 \pi} \sqrt{\frac{\epsilon_0}{\mu_0}} \left( \frac{\lambda}{d} \right)^2 \\ +&= \frac{(2 \pi \lambda^{-1})^2}{8 \pi^2} 3 \lambda^2 \\ +&= \frac{3}{2} +\end{align*} +$$ + +since $$k = 2 \pi / \lambda$$. + +To find the area, recall that the maximum power received is $$V^2 / 8 R$$. This will be equal to +the area times the Poynting vector. Thus + +$$ +\begin{align*} +\frac{V^2}{8 R} &= \langle | P | \rangle A \\ +&= \frac{1}{2} \sqrt{\frac{\epsilon_0}{\mu_0}} E^2_\text{max} A +\end{align*} +$$ + +Solving for $$A$$, + +$$ +\begin{align*} +A &= \frac{V^2}{4 R} \sqrt{\frac{\mu_0}{\epsilon_0}} \frac{1}{E^2_\text{max}} \\ +&= \frac{V^2}{4} \sqrt{\frac{\mu_0}{\epsilon_0}} \frac{1}{E^2_\text{max}} +\frac{3}{2 \pi} \sqrt{\frac{\epsilon_0}{\mu_0}} \left( \frac{\lambda}{d} \right)^2 \\ +&= \frac{3}{8 \pi} \frac{V^2}{E^2_\text{max}} \left( \frac{\lambda}{d} \right)^2 +\end{align*} +$$ + +since for an infinitesimal dipole radiator + +$$ +R = \frac{2 \pi}{3} \sqrt{\frac{\mu_0}{\epsilon_0}} \left( \frac{d}{\lambda} \right)^2 +$$ + +Now we need to relate the voltage and the maximum magnitude of the electric field. The whole point +of the antenna is to pick up oscillating electric and magnetic fields, and the presence of the +latter makes voltage path dependent. So let's integrate along our (infinitely thin) antenna. Our +antenna has an infinitesimal length $$d$$, so the field will be constant across it. Thus the +integral is just $$d$$ times the electric field, i.e. $$V = E_\text{max} d$$. As such + +$$ +A = \frac{3}{8 \pi} \lambda^2 +$$ + +Finally we see that the area gain ratio is + +$$ +\begin{align*} +\frac{A}{G} &= \frac{3}{8 \pi} \lambda^2 \frac{2}{3} \\ +&= \frac{1}{4 \pi} \lambda^2 +\end{align*} +$$