Vector potential in cavity of arbitrary shape

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SUMMARY

The discussion centers on the validity of the vector potential equation \(\int A(\lambda) A^{*}(\lambda^{'}) dV = 4 \pi c^{2} \delta_{\lambda \lambda^{'}}\) within cavities of arbitrary shapes. It is established that while the equation remains valid in such cavities, determining the functions A becomes more complex. The orthogonality of modes at different frequencies is preserved regardless of the cavity shape.

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Tianwu Zang
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we know that vector potential in a resonator satisfies the equation [tex]\int[/tex][tex]A[/tex]([tex]\lambda[/tex])[tex]A[/tex][tex]^{*}[/tex]([tex]\lambda[/tex][tex]^{'}[/tex])[tex]d[/tex][tex]V[/tex]=[tex]4[/tex][tex]\pi[/tex]c[tex]^{2}[/tex][tex]\delta_{\lambda\lambda[/tex][tex]^{'}[/tex]
So how about in cavity of arbitrary shape? Does this equation still valid?
Thanks!
 
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The equation represents the orthogonality of modes of different frequencies.
It would hold in a cavity of any shape, but the functions A would be harder to find.
 

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