diff --git a/_psets/6.md b/_psets/6.md index ac6b0280e2ca31ef2b9adc162cfbf697ee2740a1..3ef88136544385009750a70e928e62a9af558229 100644 --- a/_psets/6.md +++ b/_psets/6.md @@ -7,6 +7,9 @@ title: Problem Set 6 {:.question} Find the electric field for an infinitesimal dipole radiator. +So much math... check back later. + + ## (8.2) {:.question} @@ -14,11 +17,44 @@ What is the magnitude of the Poynting vector at a distance of 1 km from an anten 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 $$E_\text{max}$$, + +$$ +\begin{align*} +E_\text{max} &= \left( 2 \sqrt{\frac{\mu_0}{\epsilon_0}} \langle P \rangle \right)^\frac{1}{2} \\ +&= \sqrt{2 \cdot 377 \si{\ohm} \cdot \num{8e-5} \si{W/m^2}} \\ +&= 0.24 \si{W/m} +\end{align*} +$$ + ## (8.3) {:.question} For what value of $$R_\text{load}$$ is the maximum power delivered to the load in Figure 8.3? + ## (8.4) {:.question}