(Created page with "\item \[R_s =\frac{1}{\sigma\delta}\] \begin{align*} \bar{P} &= \frac{1}{2}R_s|J_s|^2 (A)\\ & = \frac{1}{2\sigma\delta}|H_y|^2(b)\\ \bar{P} &= \frac{b|H_y|^2}{2\sigma\delta} \...") |
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| − | + | a) | |
| + | <math> | ||
| + | R_s =\frac{1}{\sigma\delta}\\ | ||
\begin{align*} | \begin{align*} | ||
\bar{P} &= \frac{1}{2}R_s|J_s|^2 (A)\\ | \bar{P} &= \frac{1}{2}R_s|J_s|^2 (A)\\ | ||
& = \frac{1}{2\sigma\delta}|H_y|^2(b)\\ | & = \frac{1}{2\sigma\delta}|H_y|^2(b)\\ | ||
| − | \bar{P} &= \frac{b|H_y|^2}{2\sigma\delta} | + | \bar{P} &= \frac{b|H_y|^2}{2\sigma\delta}\\ |
\end{align*} | \end{align*} | ||
| + | \\ | ||
| + | BC: H_{1t} - \cancelto{0}{H_{2t}} = J_s \\ | ||
| + | H_{it} = J_s \\ | ||
| + | |J_s| = |H_y| \text{ where: } \bar{H} = Hx\hat{x} + Hy\hat{y} \\ | ||
| + | </math> | ||
| − | \ | + | b) |
| − | \ | + | <math> |
| − | \ | + | \alpha = \frac{1}{\delta} \\ |
| + | \delta = \frac{1}{\sqrt{\pi f \mu \sigma}} \text{ (conductor)} \\ | ||
| + | \alpha = \sqrt{\pi f \mu \sigma}\\ | ||
| + | </math> | ||
| − | + | c) | |
| − | + | <math> | |
| − | + | BCs: E_{1t} = E_{2t} = 0\\ | |
| + | assuming TEM: $E_z = 0 \\ | ||
| + | </math> | ||
| − | + | d) | |
| − | + | <math> | |
| + | assuming TEM: E_z = 0\\ | ||
| + | non-TEM: E_z \approx 0 | ||
| + | </math> | ||
| − | + | *Aside: | |
| − | + | <math> | |
| − | + | P = \frac{1}{2}IV = \frac{1}{2}I^2R \\ | |
| − | Aside: | + | I = J_s(l)\\ |
| − | + | P = \frac{1}{2}|J_s|^2(l^2)R_s\\ | |
| − | + | P = \frac{1}{2}R_s|J_s|^2A\\ | |
| − | + | P = \frac{1}{2}\int \bar{E}\cdot\bar{J}dv\\ | |
| − | + | = \frac{1}{2}\int E\cdot \big(\frac{J_s}{\delta}\big)ds\\ | |
| − | + | = \frac{1}{2}\big(\frac{J_s}{\sigma}\big)\big(\frac{J_s}{\delta}\big)(A)\\ | |
| − | \ | + | = \frac{1}{2}|J_S|^2\big(\frac{1}{\sigma\delta}\big)A\\ |
| − | + | = \frac{1}{2}R_s|J_s|^2A\\ | |
| − | + | </math> | |
| − | + | ||
| − | + | ||
| − | + | ||
Latest revision as of 19:42, 3 June 2017
a) $ R_s =\frac{1}{\sigma\delta}\\ \begin{align*} \bar{P} &= \frac{1}{2}R_s|J_s|^2 (A)\\ & = \frac{1}{2\sigma\delta}|H_y|^2(b)\\ \bar{P} &= \frac{b|H_y|^2}{2\sigma\delta}\\ \end{align*} \\ BC: H_{1t} - \cancelto{0}{H_{2t}} = J_s \\ H_{it} = J_s \\ |J_s| = |H_y| \text{ where: } \bar{H} = Hx\hat{x} + Hy\hat{y} \\ $
b) $ \alpha = \frac{1}{\delta} \\ \delta = \frac{1}{\sqrt{\pi f \mu \sigma}} \text{ (conductor)} \\ \alpha = \sqrt{\pi f \mu \sigma}\\ $
c) $ BCs: E_{1t} = E_{2t} = 0\\ assuming TEM: $E_z = 0 \\ $
d) $ assuming TEM: E_z = 0\\ non-TEM: E_z \approx 0 $
- Aside:
$ P = \frac{1}{2}IV = \frac{1}{2}I^2R \\ I = J_s(l)\\ P = \frac{1}{2}|J_s|^2(l^2)R_s\\ P = \frac{1}{2}R_s|J_s|^2A\\ P = \frac{1}{2}\int \bar{E}\cdot\bar{J}dv\\ = \frac{1}{2}\int E\cdot \big(\frac{J_s}{\delta}\big)ds\\ = \frac{1}{2}\big(\frac{J_s}{\sigma}\big)\big(\frac{J_s}{\delta}\big)(A)\\ = \frac{1}{2}|J_S|^2\big(\frac{1}{\sigma\delta}\big)A\\ = \frac{1}{2}R_s|J_s|^2A\\ $
