[[回路学第二]]

- 物質中のMaxwell方程式~
&mimetex(\mathrm{div}D = \rho);~
&mimetex(\mathrm{div}B = 0);~
&mimetex(\mathrm{rot}E = -\dot{B});~
&mimetex(\mathrm{rot}H = j + \dot{D});~
&mimetex(D = \varepsilon E);~
&mimetex(H = \frac{1}{\mu}B);
- 完全導体の境界条件~
&mimetex(E \times n = 0);~
&mimetex(B \cdot n = 0);~
nは境界面の法線ベクトル

- 矩形導波管~
&mimetex(x=0,a);、&mimetex(y=0,b);に完全導体の壁があるとして、~
特定のモードの複素電磁波を考えると、~
境界条件からモードの一般形は~
&mimetex(E_x = A(y)\sin(\kappa_x x)e^{-\omega t + \kappa_z z});~
&mimetex(E_y = B(x)\sin(\kappa_y y)e^{-\omega t + \kappa_z z});~
&mimetex(E_x = A(x)\sin(\kappa_y y)e^{-\omega t + \kappa_z z});~
&mimetex(E_y = B(y)\sin(\kappa_x x)e^{-\omega t + \kappa_z z});~
&mimetex(E_z = C\sin(\kappa_x x)\sin(\kappa_y y)e^{j(-\omega t + \kappa_z z)});~
となり、~
&mimetex(\frac{\kappa_x}{a\pi} \in \mathbb{N});、&mimetex(\frac{\kappa_y}{b\pi} \in \mathbb{N});~
&mimetex(\kappa_x A(y)\cos(\kappa_x x) + \kappa_y B(x)\cos(\kappa_y y) + j\kappa_z C = 0);~
&mimetex(A'(x)\sin(\kappa_y y) + B'(y)\sin(\kappa_x x) + j\kappa_z C\sin(\kappa_x x)\sin(\kappa_y y) = 0);~
&mimetex(\kappa_x^2 + \kappa_y^2 + \kappa_z^2 = \omega^2);
を満たす。
-- TEモード。&mimetex(E_z=0);のモード。~
&mimetex(\kappa_x A(y)\cos(\kappa_x x) + \kappa_y B(x)\cos(\kappa_y y) = 0);~
&mimetex(\frac{\kappa_x A(y)}{\cos(\kappa_y y)} + \frac{\kappa_y B(x)}{\cos(\kappa_x x)} = 0);~
&mimetex(A(y) = E_{0}\cos(\kappa_y y));~
&mimetex(B(x) = -E_{0}\cos(\kappa_x y));~
&mimetex(A'(x)\sin(\kappa_y y) + B'(x)\sin(\kappa_x x) = 0);~
&mimetex(\frac{A'(x)}{\sin(\kappa_x x)} + \frac{B'(y)}{\sin(\kappa_y y)} = 0);~
&mimetex(A(x) = \kappa_y \cos(\kappa_x x));~
&mimetex(B(y) = -\kappa_x \cos(\kappa_x y));~
なので、
&mimetex(E_x = E_{0}\sin(\kappa_x x)\cos(\kappa_y y)e^{-j\omega t + j\kappa_z z});~
&mimetex(E_y = E_{0}\cos(\kappa_x x)\sin(\kappa_y y)e^{-j\omega t + j\kappa_z z});~
&mimetex(E_x = \kappa_y \cos(\kappa_x x)\sin(\kappa_y y)e^{-j\omega t + j\kappa_z z});~
&mimetex(E_y = -\kappa_x\sin(\kappa_x x)\cos(\kappa_y y)e^{-j\omega t + j\kappa_z z});~
&mimetex(E_z = 0);~
-- TMモード。&mimetex(B_z=0);のモード。~
&mimetex(j\omega B_z = \{B'(x)\sin(\kappa_y y) - A'(y)\sin(\kappa_x x)\}e^{-j\omega t + j\kappa_z z} = 0);~
&mimetex(\frac{A'(y)}{\sin(\kappa_y y)} - \frac{B'(x)}{\sin(\kappa_x x)} = 0);~
&mimetex(A'(y) = -\kappa_x \kappa_y \E_{0}\sin(\kappa_y y));
&mimetex(B'(x) = -\kappa_x \kappa_y \E_{0}\sin(\kappa_x x));
&mimetex(A(y) = \kappa_x \E_{0}\cos(\kappa_y y));
&mimetex(B(x) = \kappa_y \E_{0}\cos(\kappa_x x));

&mimetex(j\omega B_z = \{\kappa_y B(x)\cos(\kappa_y y) - \kappa_y A(x)\sin(\kappa_y y)\}e^{-j\omega t + j\kappa_z z} = 0);~
&mimetex(\frac{\kappa_y A(x)}{\sin(\kappa_x x)} - \frac{\kappa_x B(y)}{\sin(\kappa_y y)} = 0);~
&mimetex(A(x) = \kappa_x \sin(\kappa_x x));~
&mimetex(B(y) = \kappa_y \sin(\kappa_y y));~
//&mimetex((\kappa_x^2 + \kappa_y^2)E_{0} \cos(\kappa_x x)\cos(\kappa_y y) + j\kappa_z C  \sin(\kappa_x x)\sin(\kappa_y y)= 0);~
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