Physics Help Forum What does "Solutions" mean for Maxwell & Einstein Equations?
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 May 24th 2018, 02:00 AM #11 Senior Member     Join Date: Jun 2016 Location: England Posts: 487 Play nicely boys! Ok insert should have been substitute, enough said. To go back to the original question; Solution in this circumstance is another case of what we think is a well known word, being used in a somewhat different way from what we expect. These sets of simultaneous equations can be used to model "real world" scenarios. Certain observed or imagined scenario's can be successfully modelled by the equations, in other words they are situations where sensible solutions can be found for the equations. If the situation cannot be modelled then either the situation, is impossible, or the model is wrong (or at least incomplete). The term "solution" is often used to describe the results of extrapolations from the everyday situations toward the extremes. For example Black Holes were originally noticed as an extreme extrapolation of Einstein's Equations (to model the scenario at the core of a collapsing star). The fact that such an obviously ridiculous solution could be derived from them was pointed at as a flaw in Einstein's equations. Then astronomers started finding objects that look very much like the black holes found in these (originally) purely theoretical extrapolations... __________________ ~\o/~
May 24th 2018, 03:01 AM   #12
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 Originally Posted by louarnold What does "Solutions" mean for Maxwell & Einstein Equations? I am a retired engineer studying (not in school) Maxwell's and Einstein's work at a beginner level. The concept of "solutions" to their equations comes up from time to time, and this puzzles me. So I ask: What does it mean to have a "solution" to one of these equations? What is being solved for and why? I understand this question in the context of simple algebraic equations, but not for more complex systems such as Maxwell's wave equations and those of General Relativity. Can someone explain at appropriate level for me? Can someone point me to an example of a solution with obvious rational for it? Thanks, Lou.
Maxwell's equations are rather fundamental and describe how electric and magnetic fields interact with each other. Therefore, you can use them to directly solve for things like:

1. How does the change in an electric field affect the magnetic field?
2. How does the change in a magnetic field affect an electric field?
3. What is the speed of electromagnetic waves in a medium (or vacuum)?
-> Maxwell's equations can be used to derive a wave equation PDE

There's a lot of undergraduate level questions on the internet that give examples of the above sorts of problems with more precise sets of inputs and outputs.

I don't know much about general relativity though... perhaps someone else can help with that...

Last edited by benit13; May 24th 2018 at 03:03 AM.

May 24th 2018, 06:01 AM   #13
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 Originally Posted by louarnold What does "Solutions" mean for Maxwell & Einstein Equations? I am a retired engineer studying (not in school) Maxwell's and Einstein's work at a beginner level. The concept of "solutions" to their equations comes up from time to time, and this puzzles me.
I'm a bit puzzled too. Can you give me some context to that? I say this because there's sometimes some misunderstanding of what the equations are saying. For example one of Maxwell's equations is known as Faraday's Law. It's usually written as ∇ × E = − ∂B/∂t.

But it's important to note that Maxwell unified electricity and magnetism to come up with the electromagnetic field. The electromagnetic field is a dual entity, wherein electric and magnetic fields are “better thought of as two parts of a greater whole”. The equation doesn't mean changing one field creates the other. The equals sign is an “is”. The curl of E is the time rate of change of B. Because they're two aspects of the same thing, the electromagnetic field.

May 24th 2018, 09:08 AM   #14
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 Originally Posted by Farsight I'm a bit puzzled too. Can you give me some context to that? I say this because there's sometimes some misunderstanding of what the equations are saying. For example one of Maxwell's equations is known as Faraday's Law. It's usually written as ∇ × E = − ∂B/∂t. But it's important to note that Maxwell unified electricity and magnetism to come up with the electromagnetic field. The electromagnetic field is a dual entity, wherein electric and magnetic fields are “better thought of as two parts of a greater whole”. The equation doesn't mean changing one field creates the other. The equals sign is an “is”. The curl of E is the time rate of change of B. Because they're two aspects of the same thing, the electromagnetic field.
You raise some good points.

It is interesting that you should ask about the curl because this harps back to my [post 3] edit post 2

Not only is del cross E a function it is a vector valued function.
This is another way of saying it is really 3 equations shorthanded into one.
But to put any values into it you need to know all three equations.

This is why I asked about matrices (a determinant of in this case) and functions.

The function is of course a field function which describes the (vector valued) value of the field at any point in the domain.

This is why I was making the distinction between the solution of algebraic and differential equations.

It is also important to note the difference between an identity and an equation, although most people use the = sign for both.

Maxwell's are identities.

Last edited by studiot; May 24th 2018 at 03:18 PM.

May 24th 2018, 02:49 PM   #15
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 Originally Posted by studiot But I did take issue with your use of the word 'insert
Then why didn't you just say that? There was no reason to make a thing of it. One thing people should know about me is that semantics is on the bottom of my list to think about.

When I said "insert" I had one thing in mind, i.e. the slot that differential equations have into which derivatives of the dependent variables are placed. That's all I meant. Had you asked me about it or simply stated your objection clearly I'd have explained and people could use the term they like the best.

Last edited by Pmb; May 25th 2018 at 02:06 AM.

 Tags einstein, equations, maxwell, relativity, solutions

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