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# Physics/Permeability of empty space

Question
Hello,

I am interested in electromagnets, and how to calculate their strength but I am wondering about relative permeability. I have read that when calculating an electromagnet's field strength, it is necessary to multiply the permeability of the core material by the permeability of empty space, for example 200 (iron) multiplied by 4*pi*10^-7 (empty space). This is a really basic question, but from the depth of my ignorance, I am assuming this is because the electromagnet is in empty space and therefore the permeability of empty space needs to be part of the equation. Is this correct?
Would there be any difference to the calculation of relative permeability of the electromagnet's core if the entire electromagnet were submerged in a fine powder of iron particles, which was then coated in plastic? Given that the electromagnet is surrounded by this material, would the permeability of empty space part of the equation change? Would it still be necessary to multiply the core material permeability by the permeability of empty space, or would you multiply the core material permeability by the surrounding material permeability, by the permeability of empty space?
Am I understanding this concept correctly?

Best regards,
Eddie

> the permeability of empty space needs to be part of the equation. Is this correct?
Yes
Feel free to go to the word "ANYWAY" if you want to skip the history lesson.

And this gets a little weird, so bear with me.

The "permeability of empty space" - µ(0) - is a constant that came about due to scientists in the 1800s not fully understanding (at first) the connection between electricity and magnetic fields. Indeed, in 1800, NOBODY even knew that magnetic fields EVEN EXISTED. Magnetism was viewed as an invisible fluid that flowed out of certain materials.

Andre Ampere (for whom the word "amp" is named) discovered that sending a current through two parallel wires causes a force between them; sometimes attractive, sometimes repulsive. If I(1) is the current through wire 1, I(2) is the current through wire 2, and 'r' is the distance between the parallel wires, then the force per unit length formula -- ie, the force on each meter of wire -- is given by

F(Ampere)/meter = k(Ampere) x I(1) x I(2)/r

where k(Ampere) is simply a constant to make the units work out.

It WOULD have been nice if, in the 1800s, EVERYONE had agreed to define an "amp" of current in such a way that k(Ampere) would be a nice, simple number. But that didn't happen. Scientists did pretty much that; while electrical engineers and electricians defined an amp in the way it's now used -- I guess it's less scary to hear you have 100 amps of electricity going into your house than to hear 300 BILLION stat-amps doing so.

As the mathematical understanding of the relationship between electricity and magnetism developed -- and it was found that this relationship could only be understood in terms of fields -- constants that had been used previously began to be intertwined. We still use the word that Benjamin Franklin chose to describe a device that could hold electricity -- a "battery" -- which in his day meant a group of cannons on a ship*. In the same way, we still use certain constants, like µ(0). Scientists HATE to have to re-write their formulae! Thus, Ampere's Law ended up being written as
F(Ampere)/meter = µ(0) x I(1) x I(2) / 2π x r
where 'µ(0)' is that "permeability of empty space" you are asking about.
I tried to find a historical account of HOW this happened, but was unable to do so.

ANYWAY, the way the formulae are written, the strength of the magnetic field ALWAYS includes that µ(0). Determining the size of the magnetic field at any point is done by:
1) Calculating what the size of the field would be without ANY ferromagnetic (or paramagnetic, or diamagnetic) material present. This is known as the 'H' field.
2) STARTING from what you got in (1), calculating the M-field, which is ALWAYS done by multiplying the H-field by the permeability of the material.
3) Adding the M-field to the H-field, and ending up the B-field.

* Because, if you need to know, such devices in Franklin's day resembled a group of cannons!
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