Chapter 36: Ferromagnetism – Magnetic Domains & Inductance

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Welcome back to the Deep Dive.

Today, we're getting into something pretty intense.

Ferromagnetism.

You know, the really strong magnetic stuff, iron, nickel, cobalt.

Why are it so important?

It's a fantastic question.

And yeah, their magnetic effects are just orders of magnitude stronger than, say, para or diamagnetism.

We're going to stick closely to the classic way of looking at this, really trying to untack the physics,

the core ideas, and the math behind it, but intuitively.

Right, focusing on what makes these materials tick.

So where's the starting point conceptually?

How do we even begin to quantify this?

Okay, so the best way in is an analogy we already know from electrostatics.

Remember polarization?

P, the electric dipole moment per unit volume inside a dielectric.

Yeah, it measures how aligned the electric dipoles are.

Exactly.

Well, for magnetism, we do the same thing.

We define a quantity called magnetization, M.

It's simply the magnetic dipole moment per unit volume.

So M tells us how strongly magnetized the material is and in which direction.

It's our fundamental measure.

Okay, M magnetic moment per unit volume.

But where do these magnetic moments actually come from at the atomic level?

It's not like tiny little bar magnets, is it?

No, absolutely not.

That old idea of magnetic poles or charges just doesn't hold up microscopically.

Ampere figured this out ages ago.

Magnetism comes from circulating electric currents,

tiny loops of current at the atomic scale.

Mostly electron spin, right?

Primarily, yes.

Electron spin gives rise to these intrinsic magnetic moments, these little current loops.

So if you have a chunk of iron and it's fully magnetized, M is pointing strongly in one direction.

You've got trillions of these atomic current loops all lined up.

Why don't they just cancel each other out?

Like the current on one side of an atom seems like it should be canceled by the current from the atom next to it.

Ah, good question.

And they do cancel out.

If the magnetization M is perfectly uniform throughout the material, imagine it.

If every atom is identical and aligned, the upward current on the right of one atom is perfectly balanced by the downward current on the left of its neighbor.

Inside the bulk, it's awash.

Okay, if it's uniform.

But what if it's not?

What if M changes from place to place?

That's the key.

When M is non -uniform, when it varies spatially, that perfect cancellation breaks down.

And that's where we see a macroscopic current emerge.

We call this the magnetization current density.

And the non -uniformity itself creates a real measurable current.

Precisely.

Mathematically, it turns out gel -magry is given by the curl of M.

It appears wherever M is, let's say, twisting or changing intensity across space.

I think the solenoid analogy really helps picture this.

It does.

Imagine a simple cylinder magnetized uniformly along its axis.

Inside.

Perfect cancellation, like we said.

No net current.

But think about the atoms right at the surface.

Ah, they don't have a neighbor on the outside to cancel their current loop.

Exactly.

So right at the surface, you get this uncancelled sheet of current flowing around the cylinder.

The whole thing acts just like a solenoid, a coil of wire carrying current, even though there are no wires, just aligned atomic moments.

That's gemolite action.

Well, a surface current related to it.

So non -dollar directly leads to these potentially macroscopic currents, gemagallic.

Now, this connection between M and gemagallic creates a bit of a headache when we try to use Maxwell's equations, particularly Ampere's law and its differential form.

The curl of the magnetic field B normally depends on the total current density.

Which includes both the currents we actually put in, like from a battery, the conduction currents, gemagallers, and these magnetization currents, gemagallic.

Right.

And gemag depends on M, which itself depends on the field B.

It gets circular and frankly messy to calculate things directly, especially in complex materials.

Yeah, that sounds like a nightmare for designing anything.

How do you handle that?

Well, physicists and engineers came up with a clever workaround.

They defined a new vector field called H, the magnetizing field.

Okay.

Another field.

Why?

It's defined specifically to simplify things.

The way H is constructed, its curl depends only on the conduction currents, the ones we directly control.

It sort of mathematically bundles up the effect of M and hides it away for convenience.

So B is the real physical magnetic field, the one that exerts forces.

Yes.

B is the fundamental field.

And H is like the field due to the external currents only, ignoring the material's response for a moment.

That's a great way to think about it.

The relationship is actually pretty simple algebraically.

H is basically B divided by the permeability of free space minus M or rearranging Y dollar equals mu dollar H plus M.

B depends on both the external effort related to H and the material's internal response M.

Ah, okay.

That structure by dollar equals mu dollar H plus M.

And that looks really familiar.

It reminds me strongly of electrostatics.

Exactly.

There's a deep analogy here.

Remember the electric displacement field D.

We had DO dollar, L was epsilon D, L plus PS.

Right.

Where E was the physical electric field, P was the polarization, and D was related only to the free charges we placed, ignoring the bound charges in the dielectric.

Precisely the same idea.

H is the magnetic analog of D.

Its source is the free or conduction current, DO plus.

B is the analog of E, the total physical field.

M is the analog of P, the material's internal response.

Keeping this parallel in mind really helps sort out their roles.

So if I know the wires and the current I'm putting through them, I can calculate H pretty directly.

Usually yes.

In many setups, H is determined directly by the geometry and the conduction current.

That's why it's often called the magnetizing field.

It represents the effort you're putting in with your external currents to try and magnetize the material.

And the B field you actually get depends on that effort,

H, plus how the material itself responds.

M.

You got it.

H is the input.

D is the resulting total field amplified or modified by M.

Now where ferromagnetism gets really interesting and different is when we look at how B actually behaves when we apply an H field experimentally.

For simple paramagnetic or diamagnetic materials, M is usually small and proportional to H.

So B is proportional to H.

Nice and linear.

But not in iron.

Not at all.

In ferromagnetic materials, the relationship between B and H is highly non -linear, and it even depends on the history of the material, what fields it's been exposed to before.

This is where the famous hysteresis loop comes in, right?

How do they measure that?

The classic experiment uses a toroid like a doughnut of the ferromagnetic material, say iron.

You wrap wire around it and pass a current through the wire.

Okay.

And the current in the wire directly sets up the H field inside the toroid.

We can control H precisely by controlling the current.

Exactly.

And then you measure the resulting B field inside the iron, maybe with a secondary coil or a hall probe, so you can plot B versus H.

What happens when you start with un -magnetized iron and slowly crank up the current, increasing H?

B increases, but very rapidly at first.

Much faster than H is increasing.

The iron strongly amplifies the field.

Eventually, though, the curve levels off.

B reaches a maximum value called saturation.

Saturation, meaning all the atomic magnetic moments are basically aligned as much as they possibly can be.

Pretty much, yeah.

You can't get any more magnetization out of it.

But the weird part is what happens when you decrease the current, reducing H back towards zero?

Let me guess.

B doesn't go back to zero along the same path.

Not even close.

When H gets back to zero, current off, B is still quite large.

The iron retains a significant amount of magnetization.

This remaining field is called the remanent field, Br dollars.

It's become a permanent magnet.

Wow.

So to get B back to zero, you actually have to fight against that remanence.

You do.

You have to reverse the current, apply H in the opposite direction.

The amount of negative H needed to force B down to zero is called the coercive force, or coercivity.

And if you keep cycling the current, making H go positive and negative, B traces out this characteristic loop shape.

That's the hysteresis loop.

Hysteresis is meaning lagging behind.

B always lags behind H because of this memory or history dependence in the material.

Now you mentioned this loop isn't just a curiosity.

It has real physical consequences.

Something about energy.

Absolutely critical.

It turns out the area enclosed by the BH hysteresis loop represents energy lost per unit volume during each cycle.

Lost?

Where does it go?

It's dissipated as heat within the material.

Every time you force those magnetic domains, we'll get to those, to reorient back and forth, there's internal friction, effectively, and it generates heat.

Okay, so if you're building, say, a transformer core, which gets magnetized and demagnetized 50 or 60 times a second by the AC current.

You want that loop area to be as small as possible.

Otherwise, the transformer gets incredibly hot and wastes a huge amount of energy.

That's why they use special soft magnetic materials like transformer iron, which are designed to have very narrow hysteresis loops.

But for a permanent magnet, you want the opposite, right?

You want it to hold its magnetization strongly.

Exactly.

For a permanent magnet, you want a high remnant field, dollar dollars, and a large course of force, meaning a wide hysteresis loop.

You want it to be magnetically hard, difficult to demagnetize.

So the shape of the loop tells you a lot about the material suitability for different applications.

That makes perfect sense for applications, using iron cores and inductors, for instance.

The high permeability lets you get a lot more inductance in a smaller space compared to an

Definitely.

But as we just discussed, it's for AC.

You absolutely need that soft iron with the narrow loop to minimize heat losses.

Let's look at another key application,

the electromagnet, especially one with an air gap.

Like the C -shaped magnets you see used for lifting things, maybe.

There's the iron core, but then a gap where the actual work gets done.

That's a classic case study.

Let's say you have that C -shaped core with windings creating a magnetic circuit, but there's a small air gap.

How does the magnetic field behave across that gap, and how does the H field, our effort field, distribute itself?

Okay, two main physics principles apply.

First, magnetic flux lines are continuous.

They have to loop around.

So the total magnetic flux, which is the B field times the cross -sectional area, must be the same in the iron part as it is crossing the air gap.

V times area is conserved around the circuit.

Okay, flux continuity makes sense.

Second, Ampere's law in integral form tells us that the line integral of H all the way around the magnetic circuit through the iron and across the gap is equal to the total conduction current enclosed by the loop, basically.

The number of turns times the current you put in the winding.

Right, so the total effort integral of H is fixed by the current Ni.

Now think about the relationship $1,

the Moh.

In the air gap, $1 is just $1, the permeability of free space, which is very small.

In the iron core, molar is maybe thousands of times larger than a molar dollar.

Ah,

since B has to be roughly the same in both, assuming the area is similar, if molar is huge in the iron, H must be tiny in the iron.

Exactly.

And if the molar is tiny in the gap, H must be huge in the gap to maintain the same B.

You've got it.

The total integral of H is fixed by Ni.

But because the iron is so magnetically easy, high little effort is needed to push the flux through the iron path.

The vast majority of the magnetomotive force, Ni, gets dropped across the air gap.

So almost all your driving currents effort is spent establishing the H field in that tiny air gap, not in the iron itself.

Precisely.

It's like an electric circuit with a huge resistor and a tiny resistor in series.

Almost all the voltage drops across the big resistor.

Here, the air gap is the region of high magnetic reluctance, demanding most of the H field effort.

Okay, this macroscopic behavior is fascinating, but let's go deeper.

What's the microscopic reason for ferromagnetism being so strong in the first place?

Why does iron want to magnetize so intensely, even spontaneously?

Right, we know the moments come from electron spins.

But the real puzzle is the strength of the alignment.

At room temperature, thermal energy is constantly trying to jiggle things around and randomize orientations.

For magnetism to persist, there must be some incredibly powerful cooperative effect aligning those electron spins.

Stronger than typical magnetic dipole interactions.

Way stronger.

Simple magnetic interactions between atomic moments are far too weak to explain ferromagnetism.

This led to the Weiss theory, a sort of phenomenological model.

What did Weiss propose?

He hypothesized a very strong internal magnetic field, which he called the molecular field, biomoldole.

The crucial assumption was that this field is itself proportional to the overall magnetization M of the material.

Whoa, okay, so the more magnetized the material gets, the stronger this internal aligning field becomes, which makes it even more magnetized.

It's a self -reinforcing feedback loop.

Exactly.

It's a cooperative effect.

This powerful molecular field, which we now understand arises from quantum mechanical exchange interactions.

Not classical magnetism.

Forces neighboring atomic moments to align strongly, overcoming the randomizing effect of heat.

And this leads to spontaneous magnetization below a certain temperature.

Yes, that's the Curie temperature, TCA.

Below TADO, the aligning force of the molecular field winds out over thermal energy, and the material spontaneously develops a large magnetization even with zero external field.

Above TTROL, thermal energy dominates, the alignment breaks down, and the material behaves like an ordinary paramaggot.

But wait, if iron spontaneously magnetizes below its Curie point, which is very high, why isn't every piece of iron I pick up already a strong magnet?

Ah, that's the final piece.

Magnetic domains.

The Weiss theory explains why small regions want to be fully magnetized.

But a large chunk of material can lower its overall magnetic energy by breaking up into many small regions,

called domains.

Domains?

Yeah, think of them as microscopic zones.

Within each domain, the magnetization is uniform and saturated, pointing in one specific direction, just as the Weiss theory suggests.

But in an un -magnetized piece of iron, the directions of these different domains are randomly oriented relative to each other.

Ah, so the domains are all strongly magnetized internally, but their random orientations cancel out on a large scale, giving no net external field.

Exactly.

The bulk material looks un -magnetized overall.

So when we apply an external H field to magnetize it, what are we actually doing to these domains?

We're not creating the magnetization from scratch.

It's already there, just scrambled.

The applied H field does two main things.

First,

domains that happen to be aligned favorably with the field grow larger at the expense of unfavorably aligned ones.

The domain walls move.

Second, as the field gets stronger, the magnetization vector within entire domains can rotate to align more closely with the external field.

Okay, so that initial steep rise on the BH curve isn't atoms aligning one by one.

It's mostly these domains growing and rotating until, at saturation, essentially the whole thing has become one large, aligned domain.

That's the modern picture, yes.

The domain structure is crucial for understanding the whole hysteresis process, the remnants, the coercivity.

It all comes down to how easily these domain walls move and how easily the domain magnetization rotates.

Wow.

Okay, that's quite a journey.

We started with defining magnetization M from atomic currents.

So how non -uniform M leads to magnetization currents.

Understood the need for the auxiliary field H to simplify calculations by focusing on external currents.

Distinguish B, the total physical field, from H, the magnetizing effort, drawing that parallel with E and D.

Then explored the experimental reality, the non -linear BH curve, the hysteresis loop, remnants, coercivity, and the absolutely critical concept of energy loss represented by the loop area.

And finally, touch on the microscopic origin, the incredibly strong quantum mechanical alignment force, modeled by the Weiss molecular field, leading to spontaneous magnetization below the Curie temperature, but manifesting in bulk materials through the complex structure of magnetic domains.

It really paints a picture of how complex these everyday materials actually are.

So what's the final thought for someone trying to really grasp ferromagnetism?

I think the key takeaway is that ferromagnetism sits right at the intersection of fundamental physics and material science.

The basic laws of E and M, like Maxwell's equations, are still the foundation.

But predicting the actual behavior of a piece of iron, that depends crucially on M's response to H.

And M isn't just a simple function of H, it depends on quantum mechanics,

the exchange interaction, thermodynamics, the Curie temperature, and the material's physical microstructure,

the domains, the crystal grains, impurities, stresses, its entire history.

So you can't just solve the equations, you have to understand the material itself.

Exactly.

Studying ferromagnetism forces you to appreciate that the microscopic properties we observe are an emergent result of physics acting across multiple scales, from the quantum spin of the electron up to the millimeter scale arrangement of domains.

It's a rich, complex, and incredibly useful area of physics.