在圆柱形波导管中传播的电磁波 其电磁场的横向分量(即垂直于管轴的分量)都可以用纵向分量(即平行于管轴的分

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在圆柱形波导管中传播的电磁波,其电磁场的横向分量(即垂直于管轴的分量)都可以用纵向分量(即平行于管轴的分量)表示,试求出这种表达式。

热搜:电磁波的电场强度,磁场强度和传播速度

答案解析

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  • 设圆柱形波导管中充满电容率为ε、磁导率为μ的均匀介质,其中传播的电磁波的频率为ω,则Maxwell方程组变为
       latex.action?latex=XGJpZ3RyaWFuZ2xlZG93biBcdGltZXMgRT1pXG9tZWdhXG11SA%3D%3D    ①
       latex.action?latex=XGJpZ3RyaWFuZ2xlZG93biBcdGltZXMgSD0taVxvbWVnYVxtdUU%3D    ②
       建立柱坐标系,以管的轴线为z轴,则管中传播的电磁波为
       E(r,t)=E(r)ei(kzz-ωt)    ③
       H(r,t)=H(r)ei(kzz-ωt)    ④式中Kz是传播矢量k的z分量。
       将③④式代入①式,有
       latex.action?latex=XGZyYWN7MX17XHJobyB9XGZyYWN7XHBhcnRpYWwgRV97en19e1xwYXJ0aWFsIFxwaGl9LVxmcmFj%0D%0Ae1xwYXJ0aWFsIEVfe1xwaGl9fXtccGFydGlhbCB6fT1cZnJhY3sxfXtccmhvIH1cZnJhY3tccGFy%0D%0AdGlhbCBFX3t4fX17XHBhcnRpYWwgXHBoaX0taWtfe3h9RV97XHBoaX09aVxvbWVnYVxtdUhfe1xy%0D%0AaG8gfQ%3D%3D   ⑤
       latex.action?latex=XGZyYWN7XHBhcnRpYWwgRV97XHJobyB9fXtccGFydGlhbCB6fS1cZnJhY3tccGFydGlhbCBFX3t6%0D%0AfX17XHBhcnRpYWwgXHJobyB9PWlrX3t6fUVfe1xyaG8gfS1cZnJhY3tccGFydGlhbCBFX3t4fX17%0D%0AXHBhcnRpYWwgXHJobyB9PWlcb21lZ2FcbXVIX3tccGhpfQ%3D%3D  ⑥
       latex.action?latex=XGZyYWN7MX17XHJobyB9XGZyYWN7XHBhcnRpYWwgfXtccGFydGlhbCBccmhvIH0oXHJobyBFX3tc%0D%0AcGhpfSktXGZyYWN7MX17XHJobyB9XGZyYWN7XHBhcnRpYWxFIFxyaG8gfXtccGFydGlhbCBccmhv%0D%0AIH09aVxvbWVnYVxtdUhfe3p9  ⑦
       将③④代入②式,有
       latex.action?latex=XGZyYWN7MX17XHJobyB9XGZyYWN7XHBhcnRpYWxIX3t6fSB9e1xwYXJ0aWFsIFxwaGl9LVxmcmFj%0D%0Ae1xwYXJ0aWFsSF97XHBoaX0gfXtccGFydGlhbCBfe3p9fT1cZnJhY3sxfXtccmhvIH1cZnJhY3tc%0D%0AcGFydGlhbEhfe3p9IH17XHBhcnRpYWwgXHBoaX0gLWlrX3t6fUhfe1xwaGl9PS1pXG9tZWdhXHZh%0D%0AcmVwc2lsb24gRV97XHJobyB9  ⑧
       latex.action?latex=XGZyYWN7XHBhcnRpYWxIX3tccmhvIH0gfXtccGFydGlhbCBfe3p9fS1cZnJhY3tccGFydGlhbEhf%0D%0Ae3p9IH17XHBhcnRpYWwgX3tccmhvIH19PWlrX3t6fUhfe1xyaG99,latex.action?latex=SF97XHJob30tXGZyYWN7XHBhcnRpYWxIX3t6fSB9e1xwYXJ0aWFsIF97XHJobyB9fT0taVxvbWVn%0D%0AYSBcdmFyZXBzaWxvbiBIX3tccGhpfQ%3D%3D  ⑨
       latex.action?latex=XGZyYWN7MX17XHJob31cZnJhY3tccGFydGlhbCB9e1xwYXJ0aWFsIF97XHJobyB9fShccmhvSF97%0D%0AXHBoaX0pLVxmcmFjezF9e1xyaG99XGZyYWN7XHBhcnRpYWxIX3tccmhvfSB9e1xwYXJ0aWFsIF97%0D%0AXHBoaSB9fT1pXG9tZWdhIFx2YXJlcHNpbG9uIEhfe3p9  ⑩
       从⑥和⑧式消去Hψ,得
       latex.action?latex=RV97XHJob309XGZyYWN7aX17XG9tZWdhXnsyfVx2YXJlcHNpbG9uIFxtdS1rX3t6fV57Mn19XGxl%0D%0AZnQoIGtfe3p9XGZyYWN7XHBhcnRpYWwgRV97en19e1xwYXJ0aWFsIFxyaG99K1xmcmFje1xvbWVn%0D%0AYVxtdX17XHJob31cZnJhY3tccGFydGlhbCBIX3t6fX17XHBhcnRpYWwgXHBoaX0gXHJpZ2h0KQ%3D%3D
       从⑤和⑨式消去Hρ,得
       latex.action?latex=RV97XHBoaX09XGZyYWN7aX17XG9tZWdhXnsyfVx2YXJlcHNpbG9uIFxtdS1rX3t6fV57Mn19XGxl%0D%0AZnQoIFxmcmFje2tfe3p9fXtccmhvfVxmcmFje1xwYXJ0aWFsIEVfe3p9fXtccGFydGlhbCBccGhp%0D%0AfS1cb21lZ2FcbXVcZnJhY3tccGFydGlhbCBIX3t6fX17XHBhcnRpYWwgXHJob30gXHJpZ2h0KQ%3D%3D
       从⑤和⑨式消去Eψ,得
       latex.action?latex=SF97XHJob309XGZyYWN7aX17XG9tZWdhXnsyfVx2YXJlcHNpbG9uIFxtdS1rX3t6fV57Mn19XGxl%0D%0AZnQoIGtfe3p9XGZyYWN7XHBhcnRpYWwgRV97en19e1xwYXJ0aWFsIFxyaG99LVxmcmFje1xvbWVn%0D%0AYVxtdX17XHJob31cZnJhY3tccGFydGlhbCBIX3t6fX17XHBhcnRpYWwgXHBoaX0gXHJpZ2h0KQ%3D%3D
       从⑥和⑧式消去Eρ,得
       latex.action?latex=SF97XHBoaX09XGZyYWN7aX17XG9tZWdhXnsyfVx2YXJlcHNpbG9uIFxtdS1rX3t6fV57Mn19XGxl%0D%0AZnQoIFxmcmFje2tfe3p9fXtccmhvfVxmcmFje1xwYXJ0aWFsIEhfe3p9fXtccGFydGlhbCBccGhp%0D%0AfStcb21lZ2FcbXVcZnJhY3tccGFydGlhbCBFX3t6fX17XHBhcnRpYWwgXHJob30gXHJpZ2h0KQ%3D%3D
       以上四式即为用纵向分量表达的横向分量的表达式。建立坐标系,从Maxwell方程组出发,以单色简谐波来讨论。
       [引申拓展] 对于矩型波导管,亦同样可以用纵向分量来表达横向分量,方法相同,只是无需建立柱坐标系而采用直角坐标系更简单