Timeline for When is the assumption $\nabla \cdot D = 0$ justified for a waveguide?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 16, 2017 at 7:50 | vote | accept | Kosha Misa | ||
| Jan 15, 2017 at 22:27 | comment | added | Kosha Misa | Please, let us continue in chat. | |
| Jan 15, 2017 at 21:46 | comment | added | user130529 | Let us continue this discussion in chat. | |
| Jan 15, 2017 at 18:50 | answer | added | user130529 | timeline score: 1 | |
| Jan 15, 2017 at 18:22 | comment | added | Kosha Misa | I agree, in free space it holds. But just try to some free charges (like in a cavity for example) and it wont hold. When it comes to the PEC (perfectly electric conductor) - sure, I agree that one needs a boundary condition. | |
| Jan 15, 2017 at 18:12 | comment | added | user130529 | Sorry, you are wrong, the eigenvalue equation above implies that $\nabla \cdot E = 0$ in the free space. Proof: if $ \nabla \times (\nabla \times E) = \omega^2 E $, applying div to both sides yields $ \nabla \cdot \nabla \times (\nabla \times E) = \omega^2 \nabla \cdot E $. Using $ \nabla \cdot \nabla \times A = 0$ for all field $A$ (with here $A=\nabla \times E$), we get $ 0 = \omega^2 \nabla \cdot E $, hence, if $\omega \neq 0$, we have $ 0 = \nabla \cdot E$, Q.E.D. | |
| Jan 15, 2017 at 17:37 | comment | added | Kosha Misa | I think we are writing about different things. In the eigenvalue equation from above div E must not be $0$. Simply take into account the third maxwells equation: $\nabla \cdot D = \rho $. And here I have 2 conductors, hence TEM-modes exist, hence surface charges exist, hence in a specific area $\nabla \cdot D \not= 0$. | |
| Jan 15, 2017 at 14:06 | comment | added | user130529 | The divergence of the curl of any vector field is always zero, see Vector calculus identities. Hence your divergence is zero, both in 2D (for TE modes) and 3D. Here you cannot ignore the central conductor. You need to set the same boundary condition as on the exterior conductor: the electric field tangent component must be zero. For numerical solving, you can use Nedelec's edge element H(curl). | |
| Jan 15, 2017 at 13:23 | history | edited | ACuriousMind♦ | CC BY-SA 3.0 |
grammar improvements; proper title; spelling out acronyms
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| Jan 15, 2017 at 13:08 | comment | added | Kosha Misa | curl curl A = (div grad) A - grad (div A), according to vector identities. Applying div to it doesn't necessarily end up in $0$. But again, I can't only consider a free space and the trivial solution of the eigenvalue equation is not of interest to me. | |
| Jan 15, 2017 at 12:56 | comment | added | user130529 | You have div curl curl E = 0 on the left side, so you have div E = 0 on the right side. BTW, if you are TE, then $E_z = 0$, right? (assuming $z$ is the axis of the guide). Then the 2D divergence is zero as well (when there are no charges, in particular outside the conductor). | |
| Jan 15, 2017 at 12:28 | comment | added | Kosha Misa | If I don't include the conductor, then the whole equation makes no sense. Without a conductor, the current wont flow. This must be included in the equation. And regarding the div curl - it's not that easy, since curl curl mus be calculated first. It results in a sum, which, when divergence is applied doesn't end up being $0$ | |
| Jan 15, 2017 at 12:00 | comment | added | user130529 | OK, so if you use the div curl = 0 identity in $ \nabla \times \nabla \times E(x,y,z) = \omega^2 E(x,y,z) $, don't you get $\nabla \cdot E=0$? (of course this is valid only in the free space, you can't include the conductor) | |
| Jan 15, 2017 at 11:48 | history | edited | Kosha Misa | CC BY-SA 3.0 |
3D case
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| Jan 15, 2017 at 11:42 | comment | added | user130529 | Are you taking the divergence in 3D or 2D? (in other words, are you considering a 3D vector $E(x,y,z)$ with 3 space variables or a 2D vector $E(x,y)$ with 2 space variables ?) I suggest you specify this in each equation in your question. | |
| Jan 15, 2017 at 10:13 | history | asked | Kosha Misa | CC BY-SA 3.0 |