Timeline for Separating static and time-dependent components of E and B
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Dec 17, 2019 at 18:05 | comment | added | S. McGrew | Something along the lines of what @knzhou commented, but with an explanation of why it is okay to "add an arbitrary solution to the wave equation." My intuition says it's okay, but I'm hoping for something more rigorous. It's obvious that the sum is a valid solution, but not obvious to me (yet) that the sum is indeed the general solution. | |
| Dec 17, 2019 at 18:02 | comment | added | S. McGrew | My question is not so much about physics, as about the propriety of a specific mathematical procedure. Specifically, I'd like to know if it is appropriate in the general time independent source (charge/current density) case to represent E and B as the sums of time-dependent wave field components and time-independent components, to solve for the two separately, and then to assert that the general solution (in this case) is a linear sum of the two separate solutions. "What you did is fine" is encouraging, but doesn't tell me why it's fine. A link to a math justification would serve nicely. | |
| Dec 17, 2019 at 17:44 | comment | added | Gareth Meredith | If it is the last statement, there are many instances of static fields that do not radiate away so easily. This is by our best guesses brought on by many different circumstances such as related to the zeno effect where the wave function evolution is effectively disabled in some given finite time, or an electron oscillating in the ground state will not give up radiation. electromagnetic field is a magnetic field produced by moving electrically charged objects. They have zero Hertz. | |
| Dec 17, 2019 at 17:36 | comment | added | Gareth Meredith | I'm sorry, what is the issue? I already said what you have done is fine. The reference is related to what you have stated, using slightly different notation. | |
| Dec 16, 2019 at 18:41 | comment | added | S. McGrew | Looks like that link explains Maxwell's equations in the static case, without addressing the issue I'm asking about. | |
| Dec 16, 2019 at 15:08 | comment | added | Gareth Meredith | google.com/url?sa=t&source=web&rct=j&url=http://… | |
| Dec 16, 2019 at 15:04 | comment | added | S. McGrew | I imagine you're right that it is fine, but can you provide a reference? It might be more of a math question than a physics question. | |
| Dec 16, 2019 at 14:52 | comment | added | Gareth Meredith | Yes it is fine, the only thing is you have taken a particular gauge of choice, the full relationship consists of a relativistic mixed equation, with Dirac matrices under geometric algebra, this look like ∇_μv E = ∂_μv ⋅ E + (∂_μv B/∂t)^k γ_kγ_0 = ∂_μv ⋅ E + iσ ⋅ (Γ_μv x E)^k γ_k γ_0 = ∂_μv ⋅ E + (∂Ω_μv/∂t)^k a_k | |
| Dec 16, 2019 at 13:48 | history | edited | S. McGrew |
edited tags
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| Dec 16, 2019 at 13:41 | history | edited | S. McGrew | CC BY-SA 4.0 |
emphasized a term
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| Dec 16, 2019 at 13:32 | comment | added | S. McGrew | The boundary conditions are that $ρ$ and $j⃗ $ are confined to a finite region, and that both are time-independent. @knzhou, thanks for the link to Jefimeko's equations. Am I right that E and B can *in general* be separated into time-dependent parts that satisfy the wave equations and time-independent parts that satisfy e.g. my last equation when $ρ$ and $j⃗ $ meet my conditions? | |
| Dec 16, 2019 at 13:28 | answer | added | Gareth Meredith | timeline score: -1 | |
| Dec 16, 2019 at 9:54 | comment | added | Alex Trounev | What boundary conditions do you set for these equations? | |
| Dec 16, 2019 at 6:27 | comment | added | knzhou | The idea is just fine, but trying to solve the differential equations directly is a pain. It's easier to solve for the potentials (not too hard when everything is time-independent) and then use those to get the fields. The results will be the time-independent case of Jefimenko's equations, to which you can add an arbitrary solution to the wave equation. | |
| Dec 16, 2019 at 6:14 | history | asked | S. McGrew | CC BY-SA 4.0 |