I want to solve following Maxwell's equation. $$ \triangledown ^{2}E+\frac{\omega^{2}}{c^{2}}E=0 $$ But, Electric field has x, y component in 2D geometry. So, it will be $$ \frac{\partial^{2} E_x}{\partial x^2}+\frac{\partial^{2} E_x}{\partial y^2}+\frac{\omega^{2}}{c^{2}}E_x=0 $$ $$ \frac{\partial^{2} E_y}{\partial x^2}+\frac{\partial^{2} E_y}{\partial y^2}+\frac{\omega^{2}}{c^{2}}E_y=0 $$ They are perfectly separated, so if I put a y-directional source Jy on a surface. It seems that it wouldn't affect to Ex. So, Ex in all domain would be 0, but it isn't in real. Could you tell me what is worng?
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1$\begingroup$ you also have to consider that in the absence of charges divE=0 (you have a sign error in the equation) $\endgroup$– hyportnexCommented Sep 1, 2022 at 12:50
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$\begingroup$ Thanks for your comment. I modified the sign. $\endgroup$– JinCommented Sep 1, 2022 at 12:59
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$\begingroup$ What are the boundary conditions? The equations have infinitely many solutions if you don't specify them. $\endgroup$– JavierCommented Sep 1, 2022 at 14:05
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$\begingroup$ The boundary condition is n x E = 0 on conductor surface, and source is Jy on a boundary. $\endgroup$– JinCommented Sep 1, 2022 at 14:46
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