Inductance of a Solenoid Around a Conducting Sphere?

[Who Am I]

I got into a little debate about the nature of a problem where you put a giant solenoid around the equator of Mars to give it a magnetic field (not my idea, I like futuristic things but... there are probably better things to worry about).

Anyways, I got into a debate about the effect of the core having an induced current. Someone claimed that it would take hundreds of thousands of years for it to decay. Which, to me, makes no sense since the formula they used was for skin depth, which doesn't even make sense to use here because there is no driving frequency - it's a steady state solenoid. If I'm not mistaken, skin depth is derived from a system with a constant frequency that has "leveled off," that is, that any other induction effects have roughly decayed out and now you have a constant driving frequency as if that is what was happening forever. Correct me on this if I am wrong.

In fact, I'm pretty sure that this person simply calculated the frequency you need to get a skin depth of 1800 km. Which, I'm not sure if that really even makes that much sense given the geometry of the problem.

He took this equation:


He then plugged in the skin depth as the radius of Mars's core, about 1800 km, and solved for omega, claiming that the period you get from omega is the decay time, but as 1/4th since it decays to zero. I think that this calculation is kind of silly, since it just actually tells you the angular frequency (and period) you need to get a skin depth of 1800 km, which if you're talking about a massive current running through a wire with a diameter of 50,000 miles would actually give you an adequate description of what's physically happening. He also used the vacuum permeability, and he used the resistivity of iron.

I'm pretty sure this is just not how any of this works. I'm looking for people who have solid E&M backgrounds who can solve this problem, at least a simplified version of it, with Maxwell's equations, boundary conditions and the like (which, the person who I was debating with didn't even really seem to have knowledge of).

It seemed that other users also were doing the same calculation, and it's more of an engineer's forum - so, perhaps they're misapplying a shorthand, and they're not doing anything EE related professionally? I'm not sure how physics is taught in engineering, outside of the basic physics classes that they usually take through the physics department, which teach on a much more basic level and lack rigor in comparison to the upper division E&M courses you find in physics (which I am rusty on but have taken).

So, help me out! Do these people actually know something I don't, or am I not crazy in thinking that they're full of it?

 

[Baluncore]

I would trust their prediction, they are on the right track.
Skin effect is very important, but only the longest time delays will be due to the internal parts of the 1800 km radius conductive core.

When you turn on the electromagnet with a voltage step, the rise in current will have an AC components that will penetrate Mars at the speed of EM waves in Mars. That speed will be dependent on the magnetic, electric and resistive composition of the successive layers, through the equation you provided.

Each transition in stratification will transmit some EM wave while reflecting the difference. The magnetic effect on your equatorial coil windings on the surface will be the sum of all the delayed reflections from within the body of Mars. The first reflection from the surface of the conductive core will be very significant. But it will then take a long long time for the eddy current in the core to approach a final predicted value.

 

[Who Am I]

I would trust their prediction, they are on the right track.
Skin effect is very important, but only the longest time delays will be due to the internal parts of the 1800 km radius conductive core.

When you turn on the electromagnet with a voltage step, the rise in current will have an AC components that will penetrate Mars at the speed of EM waves in Mars. That speed will be dependent on the magnetic, electric and resistive composition of the successive layers, through the equation you provided.

Each transition in stratification will transmit some EM wave while reflecting the difference. The magnetic effect on your equatorial coil windings on the surface will be the sum of all the delayed reflections from within the body of Mars. The first reflection from the surface of the conductive core will be very significant. But it will then take a long long time for the eddy current in the core to approach a final predicted value.

The skin depth is the characteristic length of attenuation (the length at which the wave attenuates by a factor of 1/e) of an EM wave in matter - including conductors. The fact that the frequencies of EM waves are so high and the conductivity (σ) of conductors is so high is why they're so impenetrable by that equation:

δ=(2/μσω)^1/2

That's how skin depth relates to EM waves. The reason that the guy in question got such an absurdly large value is that he was calculating the EM wave frequency that you'd need to get a skin depth of 1800 km. Which also has a practically non-existent energy if you're quantizing the wavefunctions.

The decay time would have more to do with mutual inductance, would it not?

Correct me if I'm wrong.

 

[Baluncore]

The decay time would have more to do with mutual inductance, would it not?

It takes as long to get the historical field out of the core volume as it took to get it in there.

The velocity of propagation into a good conductor is only a few hundred m/sec. An EM wave will penetrate into a conductor from the surface at that very low velocity. Knowing the frequency, compute the velocity, then work out the wavelength of an EM wave for that frequency in the material. Half a wavelength depth ahead will be the previous half cycle with opposite phase, diffusing and merging until the energy is lost in the blur of deep eddy currents, usually by about 3 wavelengths depth. That is why skin depth is approximately half a wavelength in the material at any frequency. Low frequencies do not have such a problem with the previous half cycle, so they will penetrate to a greater depth more efficiently.

The surface of the core is a good conductor, so the surface eddy currents will make it a really good EM mirror. It will reflect most of the energy injected at the planet's surface back out from the core surface. Only a small amount will be transmitted onwards, deeper into the core.

During the time it takes you to carry out the experiment, the volume of the core will be effectively insulated from you by it's conductive mirror surface. The short term cross-sectional area of Mars will be reduced by the inaccessibility of that volume, so the inductance will initially be lower than expected for the equatorial coil. For frequencies above about a millihertz you can model the core as a spherical conductive shell surface. For uHz and nHz frequencies the bulk current circulating in the outer core will couple with the equatorial coil and give it a higher inductance than found at higher frequencies.

 

[Who Am I]

It takes as long to get the historical field out of the core volume as it took to get it in there.

The velocity of propagation into a good conductor is only a few hundred m/sec. An EM wave will penetrate into a conductor from the surface at that very low velocity. Knowing the frequency, compute the velocity, then work out the wavelength of an EM wave for that frequency in the material. Half a wavelength depth ahead will be the previous half cycle with opposite phase, diffusing and merging until the energy is lost in the blur of deep eddy currents, usually by about 3 wavelengths depth. That is why skin depth is approximately half a wavelength in the material at any frequency. Low frequencies do not have such a problem with the previous half cycle, so they will penetrate to a greater depth more efficiently.

The surface of the core is a good conductor, so the surface eddy currents will make it a really good EM mirror. It will reflect most of the energy injected at the planet's surface back out from the core surface. Only a small amount will be transmitted onwards, deeper into the core.

And EM waves with longer wavelengths have lower energy, correct? So, either you turn it on very slowly and the energy propagated into the core is small and reflects back immediately, or you turn it on quickly and it's reflected quickly. I guess the problem is how would you solve this problem using mutual inductance?

I guess you could model your startup function as a Fourier sine series, but for the piecewise function I=0 if t<0, I=αt where α is a constant if 0<t<t_f and finally I=αt_f if t>t_f. So, it's like a flat line at 0, a small slope and then a direct current after that.

But this sounds slightly contrived. It seems more conceptually sound to me to start with induction and boundary conditions for conductors, which is where the skin depth equation comes from in the first place. This was actually what my former roommate - a grad student who has passed all his comps (the exams grad students have to take in all the main major physics subjects) - said about the problem. That is, we're using a direct current, or at least approximately direct current case to model a system with oscillations.

Sorry if I'm pushing back more - I do have a habit of not trusting things until they're demonstrated pretty clearly (and referenced) and pretty well (i.e. with equations that actually solve the problem from first principles).

During the time it takes you to carry out the experiment, the volume of the core will be effectively insulated from you by it's conductive mirror surface. The short term cross-sectional area of Mars will be reduced by the inaccessibility of that volume, so the inductance will initially be lower than expected for the equatorial coil. For frequencies above about a millihertz you can model the core as a spherical conductive shell surface. For uHz and nHz frequencies the bulk current circulating in the outer core will couple with the equatorial coil and give it a higher inductance than found at higher frequencies.

I suppose the frequencies we're talking about here are on possibly an order of 1/(10 years), if there are meaningful frequencies to actually pull out. Also, out of curiosity, what is your background in E&M?

I happen to have Jackson and Pollack's Electrodynamics and there's a PDF of it floating around, so if there are pages you could reference in there, that would be pretty great. I've been using chapter 13.3 which covered EM waves in a conductor. Also earlier in the chapter are the boundary conditions for a conductor.

I'm going for depth here, I want to fully understand the problem, if at the very least because my interest in it is getting a lot of the rust off of my E&M knowledge.

 

[Baluncore]

My background is not math and physics, it is diverse, from geology, geophysics instrumentation, electronics, software, antenna design, radio surveillance & DF. I enjoy interesting EM challenges. I could get out of my depth quickly in the math of this analysis.

If you want to study this in depth then there are a number of changes needed in the approach.

You will need to move away from rectangular coordinates and take into account spherical coordinates in the core where the circulating current gets interesting towards the centre of the planet and core.

I suspect you need to abandon the traditional frequency based skin depth approach. You are not so interested in repeating sinusoidal AC, as in the introduction of a step change at time zero, with an exponentially reducing response from a deep stratified sphere. Consider instead employing Laplace rather than Fourier analysis.

You need a few different possible EM material models of Mars with a thick crust, mantle and a part liquid metallic Fe∙Ni∙S core. Will the Curie temperature of any of the material important?

There has been quite a bit of work done on the Earth's deep structure by monitoring the response over time to external fields. Maybe that is where you should look for earlier research and models for this field.

 

[Who Am I]

You are not so interested in repeating sinusoidal AC, as in the introduction of a step change at time zero, with an exponentially reducing response from a deep stratified sphere. Consider instead employing Laplace rather than Fourier analysis.

Yep, basically this was my objection to the person I was debating with. Skin depth is for AC currents and that stuff is built off of more basic concepts like Faraday's Law. So, he basically made a giant mistake.

 

[Baluncore]

Jackson and Pollack's Electrodynamics

What is the ISBN of the version you have?

 

[Who Am I]

What is the ISBN of the version you have?

0-8053-8567-3

 

[Baluncore]

That explains it, thanks.
ISBN 0-8053-8567-3, Author; Gerald L. Pollack & Daniel R. Stump. Title; "Electromagnetism". Addison-Wesley, 2001.
As opposed to;
ISBN 9788126510948, Author; John David Jackson. Title; "Classical Electrodynamics". Wiley, 2009.