diff --git a/_psets/6.md b/_psets/6.md index 3ef88136544385009750a70e928e62a9af558229..bb1422a5e27be9cea04ddff7665ce79a4c0c4610 100644 --- a/_psets/6.md +++ b/_psets/6.md @@ -49,11 +49,31 @@ E_\text{max} &= \left( 2 \sqrt{\frac{\mu_0}{\epsilon_0}} \langle P \rangle \righ \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)$$ and the power dissipated in the load resistance is + +$$ +P = I^2 R = \frac{V^2 R_l}{(R_r + R_l)^2} +$$ + +This is maximized where its derivative is zero. + +$$ +\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)