91影视

Write the expression for the resonant frequencies for both TE and TM mode.

${\mathit{蝇}}^{\mathbf{2}}\mathbf{=}{\mathit{c}}^{\mathbf{2}}\left(k_x^2+k_y^2+k_z^2\right)$ 鈥︹�� (1)

Here, c is the speed of light and k is the wave number

Here, the value of $k_x$, $k_y$and $k_z$are given as:

role="math" localid="1657449389970" $$\begin{gathered} k_x=\frac{m蟺}{a} \\ {k}_y=\frac{{\displaystyle n蟺}}{{\displaystyle b}} \\ {k}_z=\frac{{\displaystyle I蟺}}{{\displaystyle d}} \end{gathered}$$

Substitute$k_x=\frac{m蟺}{a},k_y=\frac{n蟺}{b}$and $k_z=\frac{l蟺}{d}$in equation (1).

localid="1657450941973" ${蝇}^2=c^2\left[{\left(\frac{m蟺}{a}\right)}^2+{\left(\frac{n蟺}{b}\right)}^2+{\left(\frac{l蟺}{d}\right)}^2\right]$

$蝇=c蟺\sqrt{\left[{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2+{\left(\frac{l}{d}\right)}^2\right]}$

Write the expression for the x, y, and z components of an electric field.

$$\begin{gathered} E_x(x,y,z)=\left[A\sin \left(k_{x}x\right)+B\cos \left(k_{x}x\right)\right]\sin \left(k_{x}y\right)\sin \left(k_{z}z\right) \\ {E}_y(x,y,z)=\sin \left(k_{x}x\right)\left[C\sin \left(k_{y}y\right)+D\cos \left(k_{y}y\right)\right]\sin \left(k_{z}z\right) \\ {E}_z(x,y,z)=\sin \left(k_{x}x\right)\sin \left(k_{y}y\right)\left[E\sin \left(k_{z}z\right)+F\cos \left(k_{z}z\right)\right] \end{gathered}$$

Hence, the associated electric field will be,

$E_x=\left[\begin{array}{l}B\cos \left(k_{x}x\right)\sin \left(k_{z}z\right)\hat{x}+D\sin \left(k_{x}x\right)\cos \left(k_{y}y\right)\sin \left(k_{z}z\right)\hat{y}+\\ F\sin \left(k_{x}x\right)\sin \left(k_{y}y\right)\cos \left(k_{z}z\right)\hat{z}\end{array}\right]$

Write the expression for the x component of a magnetic field.

$B_x=-\frac{i}{蝇}\left(\frac{鈭�E_z}{鈭�y}-\frac{鈭�E_y}{鈭�z}\right)鈭�$

Substitute the value of and in the above expression.

$B_x=-\frac{i}{蝇}\left(F{k}_y\sin \left(k_{x}x\right)\cos \left(k_{y}y\right)\cos \left(k_{z}z\right)-D{k}_z\sin \left(k_{x}x\right)\cos \left(k_{y}y\right)\cos \left(k_{z}z\right)\right)$

Write the expression for the y component of a magnetic field.

$B_y=-\frac{i}{蝇}\left(\frac{鈭�E_x}{鈭�z}-\frac{鈭�E_z}{鈭�x}\right)$

Substitute the value of $E_x$and $E_z$in the above expression.

$B_y=-\frac{i}{蝇}\left(B{k}_z\cos \left(k_{x}x\right)\sin \left(k_{y}y\right)\cos \left(k_{z}z\right)-F{k}_x\cos \left(k_{x}x\right)\sin \left(k_{y}y\right)\cos \left(k_{z}z\right)\right)$

Write the expression for the z component of a magnetic field.

$B_z=-\frac{i}{蝇}\left(\frac{鈭�E_y}{鈭�x}-\frac{鈭�E_x}{鈭�y}\right)$

Substitute the value of $E_y$and $E_z$in the above expression.

$B_z=-\frac{i}{蝇}\left(D{k}_x\cos \left(k_{x}x\right)\cos \left(k_{y}y\right)\sin \left(k_{z}z\right)-B{k}_y\cos \left(k_{x}x\right)\cos \left(k_{y}y\right)\sin \left(k_{z}z\right)\right)$

Hence, the associated magnetic field will be,

$B=\left[\begin{array}{l}-\frac{i}{蝇}\left(F{k}_y-D{k}_z\right)\sin \left(k_{x}x\right)\cos \left(k_{y}y\right)\cos \left(k_{z}z\right)\hat{x}-\\ \frac{i}{蝇}\left(B{k}_z-F{k}_x\right)\cos \left(k_{x}x\right)\sin \left(k_{y}y\right)\cos \left(k_{z}z\right)\hat{y}-\\ \frac{i}{蝇}\left(D{k}_x-B{k}_y\right)\cos \left(k_{x}x\right)\cos \left(k_{y}y\right)\sin \left(k_{z}z\right)\hat{z}\end{array}\right]$

Therefore, the resonant frequencies for both TE and TM modes are

$蝇=c蟺\sqrt{\left[{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2+{\left(\frac{l}{d}\right)}^2\right]}$,the associated electric field is

$E_x=\left[\begin{array}{l}B\cos \left(k_{x}x\right)\sin \left(k_{z}z\right)\hat{x}+D\sin \left(k_{x}x\right)\cos \left(k_{y}y\right)\sin \left(k_{z}z\right)\hat{y}+\\ F\sin \left(k_{x}x\right)\sin \left(k_{y}y\right)\cos \left(k_{z}z\right)\hat{z}\end{array}\right]$,and the associated magnetic field is$B=\left[\begin{array}{l}-\frac{i}{蝇}\left(F{k}_y-D{k}_z\right)\sin \left(k_{x}x\right)\cos \left(k_{y}y\right)\cos \left(k_{z}z\right)\hat{x}-\\ \frac{i}{蝇}\left(B{k}_z-F{k}_x\right)\cos \left(k_{x}x\right)\sin \left(k_{y}y\right)\cos \left(k_{z}z\right)\hat{y}-\\ \frac{i}{蝇}\left(D{k}_x-B{k}_y\right)\cos \left(k_{x}x\right)\cos \left(k_{y}y\right)\sin \left(k_{z}z\right)\hat{z}\end{array}\right]$