- #1
danielakkerma
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Hello all!
Consider a cylindrical cavity with length "d" and radius "a". Find the corresponding electric field, and the dispersion relation therein.
Maxwell's equations.
I tried to solve the appropriate vector Helmholtz equation(obtained by assuming harmonic time-dependence of the waves in the cavity).
Within the cylinder, one arrives at(where: [itex]\vec{E}=\vec{E_0}(r, \varphi, z)e^{-i \omega t}[/itex])
[tex] \vec{\nabla}^2 \vec{E_0} = \frac{\omega^2}{c^2} \vec{E_0}[/tex]
However, since [itex]\vec{E_0} = E_r \hat{r} + E_\varphi \hat{\varphi} + E_z \hat{z}[/itex], the Laplacian becomes far more convoluted in the cylindrical form.
Solving for Ez is not that difficult(with appropriate separation of variables).
However, how does one find E_r, E_phi?
After all, solving for the wave equation for the r, phi components involves:
[tex] (\vec{\nabla}^2 \vec{E_0})_r = \vec{\nabla}^2 E_r - \frac{1}{r^2}(E_r + 2\frac{\partial E_{\varphi}}{\partial \varphi})[/tex]
However, this equation for E_r requires E_phi; and obviously, the components of E_r are not necessarily identical to E_phi, so substituting one for the other(through analogous separation of variables for both) is impossible here.
Without making any other assumptions(or simplifications; for instance, I managed to simplify the problem greatly if I assumed complete azimuthal symmetry(i.e. E_phi = 0 & d/dphi =0)), is there any way to obtain an exact solution for this? Where should I turn to, next?
Thank you for your attention,
Reliant on your help,
Daniel
Homework Statement
Consider a cylindrical cavity with length "d" and radius "a". Find the corresponding electric field, and the dispersion relation therein.
Homework Equations
Maxwell's equations.
The Attempt at a Solution
I tried to solve the appropriate vector Helmholtz equation(obtained by assuming harmonic time-dependence of the waves in the cavity).
Within the cylinder, one arrives at(where: [itex]\vec{E}=\vec{E_0}(r, \varphi, z)e^{-i \omega t}[/itex])
[tex] \vec{\nabla}^2 \vec{E_0} = \frac{\omega^2}{c^2} \vec{E_0}[/tex]
However, since [itex]\vec{E_0} = E_r \hat{r} + E_\varphi \hat{\varphi} + E_z \hat{z}[/itex], the Laplacian becomes far more convoluted in the cylindrical form.
Solving for Ez is not that difficult(with appropriate separation of variables).
However, how does one find E_r, E_phi?
After all, solving for the wave equation for the r, phi components involves:
[tex] (\vec{\nabla}^2 \vec{E_0})_r = \vec{\nabla}^2 E_r - \frac{1}{r^2}(E_r + 2\frac{\partial E_{\varphi}}{\partial \varphi})[/tex]
However, this equation for E_r requires E_phi; and obviously, the components of E_r are not necessarily identical to E_phi, so substituting one for the other(through analogous separation of variables for both) is impossible here.
Without making any other assumptions(or simplifications; for instance, I managed to simplify the problem greatly if I assumed complete azimuthal symmetry(i.e. E_phi = 0 & d/dphi =0)), is there any way to obtain an exact solution for this? Where should I turn to, next?
Thank you for your attention,
Reliant on your help,
Daniel