Deriving capacitor and inductor models from Maxwell's eqs

[Allan Davis]

I'm sure the inductor model, i.e. vL(t) = iL'(t)*L follows without directly from Faraday's eq. But even there, with Faraday's equation we think of the changing magnetic field as inducing the voltage in the loop, where in the model it seems the other way around, that is, the voltage increases the current which increases the magnetic field.

And, I'm pretty sure Ampere's equation leads to the capacitor model, i.e., vC'(t) = iC(t) / C. That is, I'm sure Ampere's equation gives the E field strength between the capacitor plates when the circuit is open, but when the circuit is closed I don't see how Ampere's equation leads to the electric field acting through the wire and moving the electrons.

So, I'm asking about how to think about these things to better understand them.

 

[Simon Bridge]

In the first you "it seems" needs to be revisited: it does not matter how you intuit something, all the equation tells you is that something is related to something else. I'm sure you can use algebra to rearrange the equation so the seeming is otherwise.

In the second case, your initial model fails to include the wire ... add the wire into the model and apply maxwells equations to the resulting setup.

 

[Allan Davis]

In the first you "it seems" needs to be revisited: it does not matter how you intuit something, all the equation tells you is that something is related to something else. I'm sure you can use algebra to rearrange the equation so the seeming is otherwise.

In the second case, your initial model fails to include the wire ... add the wire into the model and apply maxwells equations to the resulting setup.

I'm familiar with Maxwell's eqs as field equation, i.e. how they describe electromagnetic waves in space, but I don't see how to apply them to a wire, for example a wire connecting the two terminals of a capacitor. When the circuit is open there is an E field pulling between the electrons and the positrons on the opposite plates, but only there (?), so when the switch is closed and say a wire and a resistor connect the capacitor terminals, where is the E field? Does it appear in the wire? What is the equation for it? How does the length of the wire figure in? Etc. I don't have a clue here.

 

[Simon Bridge]

I'm familiar with Maxwell's eqs as field equation, i.e. how they describe electromagnetic waves in space, but I don't see how to apply them to a wire, for example a wire connecting the two terminals of a capacitor.

Just the same way. You can have an electric field through a wire, that is how electronics work.

When the circuit is open there is an E field pulling between the electrons and the positrons on the opposite plates, but only there (?),

Well, there are no positrons in the plates - at least, not for long ... positrons are antimatter electrons.
Probably the best way to apply Maxwell here would be to adopt a fluid model for charge ... so the positive and negative charges in the plates are attracted to each other: well done.

...so when the switch is closed and say a wire and a resistor connect the capacitor terminals, where is the E field? Does it appear in the wire?

The E field is, initially, everywhere ... some of it is in the wire.

What is the equation for it? How does the length of the wire figure in? Etc. I don't have a clue here.

A wire with a potential difference V across length L has constant E = V/L (pointing along it) to a very good approximation. It is technically possible to work this out from Maxwell's equations.

Note: modelling electrodynamics, in detail, from scratch is very hard.

 

[Allan Davis]

Just the same way. You can have an electric field through a wire, that is how electronics work.
Well, there are no positrons in the plates - at least, not for long ... positrons are antimatter electrons.
Probably the best way to apply Maxwell here would be to adopt a fluid model for charge ... so the positive and negative charges in the plates are attracted to each other: well done.
The E field is, initially, everywhere ... some of it is in the wire.
A wire with a potential difference V across length L has constant E = V/L (pointing along it) to a very good approximation. It is technically possible to work this out from Maxwell's equations.

Note: modelling electrodynamics, in detail, from scratch is very hard.

Positive ions instead of positrons?

"A wire with a potential difference V across length L has constant E = V/L"

This is the thing I've been missing, I've been viewing E fields as emanating as straight lines out from charged particles, not as snaking along a wire. But with the E field in the wire current intuitively follows. But, which equation applies? The only equation where current appears is Ampere's Circuital Law, where it's associated with a spatial derivatives of the magnetic field and temporal derivatives of the E field ... so with a loop with a capacitor discharging through a wire there is changing magnetic field in inside the loop, probably insignificant, and a decreasing E field ... and we have to figure in the resistance of a resistor in the loop ... so I feel like I've made a start ... but I'd sure like to see an example of that calculation (probably trivial once understood- !) !

"Note: modelling electrodynamics, in detail, from scratch is very hard."

I've been working on this, and starting with Maxwell's equations in differential form (Faraday's eq and Ampere's ciricuital law) you can use the finite difference method (which is just Euler's method, which is just the formula distance = velocoty * time) right out of the box to simulate simple models, like a traveling 2-d wave, a wave hitting an obstacle, etc ... (all the papers on the subject use the Yee algorithm which is a simple mod to the FDM but it generates a lot of new notation).

 

[Simon Bridge]

Positive ions instead of positrons?

no. Just charge or "charge distribution"... not "charges", not "ions", you are using a model here, you don't care how the charge arises physically, you are just modelling it.

"A wire with a potential difference V across length L has constant E = V/L"

This is the thing I've been missing, I've been viewing E fields as emanating as straight lines out from charged particles, not as snaking along a wire.

Its the same thing... the advantage of using fields is you don't care how they are generated. Consider an electromagnetic field in space: where are the charges for the field lines to eminate from?

But with the E field in the wire current intuitively follows. But, which equation applies?

...all of them. You could try a simpler model of just having a wire with an initailly separated charge at each end but neutral in the middle.

... so with a loop with a capacitor discharging through a wire there is changing magnetic field in inside the loop,...

... the magnetic field goes around the wire, there is also a transient within the wire as the current builds. But putting these in at the start is assuming the solution.

probably insignificant, and a decreasing E field ... and we have to figure in the resistance of a resistor in the loop ... so I feel like I've made a start ... but I'd sure like to see an example of that calculation (probably trivial once understood- !) !

... not usually no. You probably won't want to model ideal components here... the resistance is distributed along the wire. If resistance were zero, the current would be infinite... but we would usually use conductivity rather than resistance.

Have you tried deriving Ohm's law from Maxwells eq?

"Note: modelling electrodynamics, in detail, from scratch is very hard."

I've been working on this, and starting with Maxwell's equations in differential form (Faraday's eq and Ampere's ciricuital law) you can use the finite difference method (which is just Euler's method, which is just the formula distance = velocoty * time) right out of the box to simulate simple models, like a traveling 2-d wave, a wave hitting an obstacle, etc ... (all the papers on the subject use the Yee algorithm which is a simple mod to the FDM but it generates a lot of new notation).

... and yet you seem to lack junior high level electronics like the form of the electric field in a wire and the fluid model for charge; or more advanced: EM boundary conditions for a conductor.

 

[Allan Davis]

Have you tried deriving Ohm's law from Maxwells eq?

No, can you give me a clue? I've been working with Maxwell's equations for a while (as a hobby :) actually I started by looking for applications of the FDM ), and never saw a derivation of the component models, and couldn't imagine how to even get started.

Come to think of it, I should try my usual method ... try to google up a derivation. I'm going to try that.

Great Scott - I'm led to Drude's theory and bouncing electrons ... not what I'm looking for. I'm wondering if there is a easier/simpler derivation. Or maybe that is the easy derivation ?

https://en.wikipedia.org/wiki/Drude_model

"and yet you seem to lack junior high level electronics like the form of the electric field in a wire and the fluid model for charge; or more advanced: EM boundary conditions for a conductor."

Yes and no. This is what happens when you learn by watching YouTube vids, and I started top down, i.e. looking at applications of the FDM. I do understand simple EM boundary conditions, e.g. perfect magnetic conductors or perfect electric conductors, as least to the extent required to model them in FDM sims.

 

[Simon Bridge]

You need to go over the basics.