how that the mode TE00 cannot occur in a rectangular wave guide. [Hint: In this case ω/c = k, so Eqs. 9.180 are indeterminate, and you must go back to Eq. 9.179. Show that B_{z} is a constant, and hence—applying Faraday’s law in integral form to a cross section—that B_{z}=0 , so this would be a TEM mode.]
\left. \begin{matrix} \text { (i) } E_{x}=\frac{i}{(\omega / c)^{2}-k^{2}}\left(k \frac{\partial E_{z}}{\partial x}+\omega \frac{\partial B_{z}}{\partial y}\right) \text {, } \\ \text { (ii) } E_{y}=\frac{i}{(\omega / c)^{2}-k^{2}}\left(k \frac{\partial E_{z}}{\partial y}-\omega \frac{\partial B_{z}}{\partial x}\right), \\ \text { (iii) } B_{x}=\frac{i}{(\omega / c)^{2}-k^{2}}\left(k \frac{\partial B_{z}}{\partial x}-\frac{\omega}{c^{2}} \frac{\partial E_{z}}{\partial y}\right), \\ \text { (iv) } B_{y}=\frac{i}{(\omega / c)^{2}-k^{2}}\left(k \frac{\partial B_{z}}{\partial y}+\frac{\omega}{c^{2}} \frac{\partial E_{z}}{\partial x}\right) \text {. } \end{matrix} \right\} (9.180)
\left. \begin{matrix} \text { (i) } \frac{\partial E_{y}}{\partial x}-\frac{\partial E_{x}}{\partial y}=i \omega B_{z}, & \text { (iv) } \frac{\partial B_{y}}{\partial x}-\frac{\partial B_{x}}{\partial y}=-\frac{i \omega}{c^{2}} E_{z}, \\ \text { (ii) } \frac{\partial E_{z}}{\partial y}-i k E_{y}=i \omega B_{x} \text {, } & \text { (v) } \frac{\partial B_{z}}{\partial y}-i k B_{y}=-\frac{i \omega}{c^{2}} E_{x}, \\ \text { (iii) } i k E_{x}-\frac{\partial E_{z}}{\partial x}=i \omega B_{y}, & \text { (vi) } i k B_{x}-\frac{\partial B_{z}}{\partial x}=-\frac{i \omega}{c^{2}} E_{y}. \end{matrix} \right\} (9.179)