Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Figures 7.2.2, 7.2.3 and 7.2.4 show the real fields at the instant $t = 0$ . They follow from the phasors (7.2.12), (7.2.23) and (7.2.28) for the lowest modes, $n = 1$ and $n = 0$ , with a real and positive amplitude $C$ (or $A_{0}$ ), by multiplying with $e^{j ω t}$ and taking the real part. Since $Re {j e^{j (ω t - τ z)}} = - sin (ω t - τ z)$ ,

TEM: TM_{1} : TE_{1} : E_{y} = - η A_{0} cos (ω t - k z), H_{x} = A_{0} cos (ω t - k z), E_{z} = C sin \frac{π y}{b} cos (ω t - τ z), E_{y} = \frac{τ}{h} C cos \frac{π y}{b} sin (ω t - τ z), H_{x} = - \frac{ω ε}{h} C cos \frac{π y}{b} sin (ω t - τ z), H_{z} = C cos \frac{π y}{b} cos (ω t - τ z), H_{y} = - \frac{τ}{h} C sin \frac{π y}{b} sin (ω t - τ z), E_{x} = - \frac{ω μ}{h} C sin \frac{π y}{b} sin (ω t - τ z) . (7.2.31)

TEM: uniform across the gap. The $E$ lines run straight from plate to plate.
TM $_{1}$ : $E$ arches between charges on one plate, $H$ is largest at the plates.
TE $_{1}$ : $H$ forms closed loops, $E$ (along $\pm x$ ) vanishes on the plates.

7.2.6 Field patterns across the gap