# Magnetic flux conservation

Hi, initially I have read that magnetic flux is conserved in iron core, also I know divergence of B is zero but this conservation implies that there is no flux at the interface of air and iron core surface. We still magnetic field even at the interface so there must be flux. Why do we exclude this part?

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If the plus and minus poles of a magnet are brought together with a small gap, the ## B ## is continuous through the air across the gap. ## \\ ## Meanwhile, for cases like a square transformer geometry, the approximation can be made that the flux stays entirely within the square iron transformer. ## \\ ## You might also find this recent discussion of interest: https://www.physicsforums.com/threa...a-magnetic-circuit.955585/page-2#post-6059155

jim hardy
If the plus and minus poles of a magnet are brought together with a small gap, the ## B ## is continuous through the air across the gap. ## \\ ## Meanwhile, for cases like a square transformer geometry, the approximation can be made that the flux stays entirely within the square iron transformer. ## \\ ## You might also find this recent discussion of interest: https://www.physicsforums.com/threa...a-magnetic-circuit.955585/page-2#post-6059155
Thanks @Charles Link, but I would like to see the proof of approximation that the magnetic flux from surface of iron core to air is negligible. I can not imagine that it is negligible because the magnetic field is not very small. Could you help me about that?

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There will be some leakage flux. If you look at post 16 of the "link" that I gave above, and a publushed paper that is shown there that describes this in detail, there is actually a possibility that this leakage flux is necessary to explain how the Poynting vector operates in regard to power transfer in the transformer. ## \\ ## It can't be proven that it is zero, because it isn't.

jim hardy
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I would like to see the proof of approximation that the magnetic flux from surface of iron core to air is negligible. I can not imagine that it is negligible because the magnetic field is not very small. Could you help me about that?

What is the direction of this magnetic field that you say is not very small? If it's parallel to the axis of the core then there is no flux through the surface you mentioned.

What is the direction of this magnetic field that you say is not very small? If it's parallel to the axis of the core then there is no flux through the surface you mentioned.
I try to say there are of course some places normal magnetic field at the interface between air and iron core is not small...

@Charles Link I would like to express that I really want to dig valuable things related to magnetization,transformers, leakage flux, surface currents, volume currents of iron core, flux conservations in core... out of a nice book or lecture.Regarding your unique expertise on magnetism could you suggest me a nice book or lecture which also includes nice proofs related to topics I mention above?

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@Charles Link I would like to express that I really want to dig valuable things related to magnetization,transformers, leakage flux, surface currents, volume currents of iron core, flux conservations in core... out of a nice book or lecture.Regarding your unique expertise on magnetism could you suggest me a nice book or lecture which also includes nice proofs related to topics I mention above?
There may be a good book out there on this material, but I have yet to see it. In college 1975-1980, we were taught the magnetic pole method, both in a book by Pugh and Pugh, which was a very hard book to learn from, as well as J.D. Jackson's book in graduate school. In 4 advanced E&M courses, I worked extremely hard, but it wasn't until 35+ years later (2010-2012), when I was able to tie the pole method to the surface current method with my own independent calculations, that this material made sense. The surface current method, which is now presented in Griffiths' E&M textbook, was shown to us at that time as an alternative theory that most likely gives a similar result as the pole method. ## \\ ## I still had not seen the MMF equation or the transformer with a gap until I saw it on Physics Forums in 2016. With the calculations I had done with the pole method and surface currents from 2010-2012, the MMF equation followed simply from a modified form of Ampere's law. ## \\ ## Perhaps @jim hardy can recommend a textbook for this topic.

jim hardy
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@Charles Link I would like to express that I really want to dig valuable things related to magnetization,transformers, leakage flux, surface currents, volume currents of iron core, flux conservations in core... out of a nice book or lecture.Regarding your unique expertise on magnetism could you suggest me a nice book or lecture which also includes nice proofs related to topics I mention above?

My textbook dates from 1950's and i'll post its name if i find it.
Look for a 1940's book ,
/start cynicism/
because they predate "Publish or Perish"
which shifted the purpose of textbooks from teaching undergrads how to do practical work
to
impressing one's academic reviewers with one's vector calculus prowess.
/end cynicism/

Thanks @Charles Link, but I would like to see the proof of approximation that the magnetic flux from surface of iron core to air is negligible. I can not imagine that it is negligible because the magnetic field is not very small. Could you help me about that?
Referring back to that long tome i wrote in the old thread @Charles Link mentioned, https://www.physicsforums.com/threa...we-apply-the-biot-savart.927681/#post-5979758
................................................................................................................................................................

Leakage inductance is present because of flux that doesn't couple both windings..
From the Wiki transformer page:

i just drew in a few loops of leakage flux..
-------------------------------------------------------------------------
Thought steps required to accept the premise that MMF and hence leakage flux is small:
..........
When there's no secondary current
the only MMF present is from primary magnetizing current
and that MMF will be small
because it's just Flux/Reluctance( of core)
and that's why in a power transformer magnetizing current is generally considered negligible . It is made by design small compared to rated load current in order to make the transformer efficient.

So leakage flux at no load will be roughly 1/(relative permeability of core) X (flux in the core)
and with relative permeability probably at least several hundred if not several thousand,
leakage flux to slide rule accuracy will be zero .
With more digits it'll round off to maybe as much as 1% of core flux which is still near zero.

So how does MMF ever get large ?

-------------------------------------------------------
Initial Refinement to that thought process

Okay let us add secondary current.
As explained in the tome referenced above,
primary MMF increases because of secondary amp-turns.
Okay,
Leakage flux will increase in same proportion ,
but core flux remains the same as before, (voltage is the same)
so leakage flux becomes a larger fraction of core flux than it was at no load.
and as you stated leakage out into the air is not any longer so small.
Designers can control by how much leakage flux increases with load to give the transformer its desired impedance .

---------------------------------------------------------------------------------------------

I had the good fortune back in early 1970's to have a three legged transformer core two feet tall and a Variac.
I wound enough turns on it to support 120 VAC and played for days trying to teach mysef to visualize the magnetic flux.
To that end i took an old solenoid coil perhaps the size of a pill bottle and used it for a flux detector. An open circuited coil will deliver volts equal proportional to dΦ/dt and when dealing with sine waves you can connect it to a high impedance AC millivoltmeter to make yourself a flux detector.
I wound a primary on one leg of that core so i could excite it to known volts/turn, , squeezed flux around it , shifting flux between the secondary legs by shorting them with just a single turn.
I drove the core into mild saturation by applying too much voltage and watched primary current ammeter peg high, and saw leakage flux increase with my solenoid coil flux detector

That's how I developed my mental model for the magnetic circuit approach . Small wonder it's how i think of magnetics to this day. But i had to see and feel it, probably you are academic enough to do it with thought experiments.

old jim

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jim hardy
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I have a copy of this book, 1901 edition, rescued from a library suffering the wrecking ball. It's delightful but the explanations are verbose .

https://www.amazon.com/dp/B006RT9IEM/?tag=pfamazon01-20

You can preview an earlier edition here
https://archive.org/details/dynamoelectricma00thomrich/page/n9

and i think there's a 1902 edition also in print.
To their credit, India seems to be the country reviving these practical older works.

old jim
@jim hardy I haven't had the chance to read much of the book yet, but the sketches in the book of the different apparatuses are quite amazing. They are done by hand, rather than by computer-aided drawing packages, and they really put a lot of work into them. ## \\ ## Edit: At 63, I'm too old to start a career working for the electrical utility companies, but the knowledge of some of the material in a book or two like the one above, along with a good E&M background, would be a good start for someone who is just entering the job market.

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jim hardy
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I rescued my copy of the 1901 edition about 1968 ..
Like you i admired the elaborate drawings, some of them fold out to about 14 inches.

I kept that book on my shelf for my whole 30+ years in the plant , referring to it several times a year . Tesla and Steinmetz are mentioned in present tense for they were contemporaries of the author.
I found his descriptions of magnetics inside machinery helpful . Of course the units have changed over the years...

His description of "Retardation of Magnetism" in railway dynamos was a breakthrough for me in figuring out a problem with our control rod position indicating system.

There is value i think in retracing the steps of our predecessors. It helps us develop similar analytical thinking skills.

old jim

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I rescued my copy of the 1901 edition about 1968 ..
Like you i admired the elaborate drawings, some of them fold out to about 14 inches.

I kept that book on my shelf for my whole 30+ years in the plant , referring to it several times a year . Tesla and Steinmetz are mentioned in present tense for they were contemporaries of the author.
I found his descriptions of magnetics inside machinery helpful . Of course the units have changed over the years...

His description of "Retardation of Magnetism" in railway dynamos was a breakthrough for me in figuring out a problem with our control rod position indicating system.

There is value i think in retracing the steps of our predecessors. It helps us develop similar analytical thinking skills.

old jim
I see that Steinmetz is apparently very important in the electrical engineering/electromagnetics area. My first encounter with that name was in regards to this mathematical problem, in the following "link", known as a Steinmetz Solid. I actually solved it in 1977, (it had been presented to our high school calculus class in 1973), but it wan't until about 2003 that a colleague at my workplace told me it is a rather well-known problem, attributed to Steinmetz. See: https://www.physicsforums.com/threads/math-challenge-by-charles-link-1.907510/#post-5716117 (It's not my intention to send this thread off on a tangent, but I think this might be of interest).

jim hardy