Chapter 28: Sources of Magnetic Field

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Welcome back to the Deep Dive, everybody.

We're diving into some pretty fundamental physics today.

We are.

It's all about magnetism, or more precisely, where magnetic fields come from.

Right, we all kind of know what magnets do, like how they attract or repel each other.

Yeah, we've seen the effects, but have you ever stopped to think, like, where does that invisible force, that magnetic field, actually originate?

I admit I haven't given it much thought.

It just sort of is, but obviously there's some underlying physics there that explains it.

Oh yeah, there's a whole world of fascinating physics behind those magnetic fields, and today we're going to unpack it all, starting from the tiniest sources, like single moving charges.

Okay, so like down to the electron level.

Exactly, and then we'll work our way up, seeing how currents and wires create these fields, and even how the very materials themselves can be sources of magnetism.

Sounds like a plan.

Let's get into it.

So first things first, how does a single tiny moving charge create a magnetic field?

Well, it's kind of mind -blowing, but anytime you've got an electric charge, like an electron, and it's moving, it inherently creates a magnetic field around it.

Just by moving.

Just by moving, and the strength of that field, how powerful it is, we call that its magnitude, and there's this equation that describes it.

b equals rho over 4 pi times qv sine phi all over r squared.

Okay, there's some symbols in there I recognize, like knee, that's that magnetic constant.

Right, and then q is the amount of charge we're talking about, and v is its velocity, how fast it's zipping around.

So a faster charge means a stronger field.

Makes sense, but what about that sine phi and the r squared in there?

So the r squared is familiar.

That tells us that the magnetic field gets weaker the farther you get from the charge, like with the square of the distance, kind of like gravity, or the electric field, they weaken with distance too.

Right, the influence fades as you move away.

Makes sense, but what about that sine phi term?

What's that telling us?

That's where things get a bit more subtle.

It's not just about how fast the charge is moving, but also the direction it's moving relative to the point where we're measuring the field.

Okay, so direction matters too.

Big time.

So phi phi is the angle between the direction the charge is moving, its velocity vector, and a line connecting the charge to where we're measuring the field.

Got it, so the angle between its path and the line to our measuring point.

Precisely, and the sine of that angle tells us how much of the charge's motion is contributing to the magnetic field at that point.

So if the charge is moving directly at our measuring point,

is the field the strongest?

Actually, it's the opposite.

If it's heading straight for us, the angle is zero and the sine of zero is zero.

No magnetic field at all.

Oh, interesting.

So what direction does create the strongest field then?

The field is actually strongest when the charge is moving perpendicular at a right angle to that line connecting it to the measuring point.

That's when phi is 90 degrees and the sine is one, its maximum value.

Okay, so a sideways motion relative to the measuring point makes the strongest field.

That's pretty cool.

It is, and it's all tied to the vector nature of magnetic fields.

Right, because magnetic fields have both a strength, the magnitude we talked about, and a direction.

Exactly.

In the capture both of those, we use a vector form of the equation.

B equals E over 4 pi times QV cross product R hat all over R squared.

Okay, cross product.

That sounds like vector math.

It is.

It's a way of combining the velocity vector V and the unit vector pointing from the charge to the field point R hat to get a new vector B that's perpendicular to both of them.

So the resulting magnetic field vector points in a direction that's perpendicular to both the charge's motion and the line to our measuring point.

Exactly.

It's a three -dimensional relationship.

I kind of get it, but it's hard to visualize those vectors in space.

Well, that's where the right -hand rule, I remember learning about that.

It's a lifesaver.

So imagine you point your right thumb in the direction the positive charge is moving, your velocity.

Okay, thumb for positive charge motion.

And then point your fingers in the direction of that R hat vector from the charge to where you want to find the field.

Got it.

Thumb for velocity, fingers for R hat.

Now the direction your palm would push, that's the direction of the magnetic field.

So the palm push gives us the B direction.

That's a really neat way to figure it out thanks to our handy right hand.

It is.

The right -hand rule pops up all over the place in magnetism, so it's a good one to remember.

Now, let's say we have more than one moving charge.

Do their magnetic fields all just add up?

Yeah, pretty much.

Just like with electric fields, we can use the principle of superposition.

Okay, superposition, that rings a bell.

Remind me what that means.

Superposition means that the total magnetic field at any point is just the vector sum of the fields from all the individual moving charges.

So we calculate each one separately,

then add them all up, taking their directions into account.

Exactly.

And that gives us the overall magnetic field created by all those moving charges together.

Makes sense.

So we've been talking about single charges,

but what about something a bit more practical, like an electric current flowing through a wire?

Well, an electric current is basically a whole bunch of charged particles, usually electrons, all moving together in the same direction.

Right, like a river of electrons flowing through the wire.

Exactly.

So to figure out the magnetic field created by a current, we can break it down into tiny little pieces of the wire, each carrying a tiny current.

So like zooming in on a tiny segment of the wire.

Precisely.

We call that tiny piece a current element.

We represent it with idl, where i is the current, and i is a vector pointing along a tiny length of the wire in the direction of the current.

Okay, so we're looking at a little piece of the current, and we want to know the magnetic field it creates.

How do we do that?

Well, there's a super important law called the law of Biot -Savart, or Biot -Savart law, that tells us exactly how to do that.

It says the tiny magnetic field created by that little piece of current, we call it dB, is equal to, well, AO over 4, again, times idls and phi over r squared.

Wait, that looks almost like the equation for a single moving charge.

It does, doesn't it?

And that's a really cool connection.

So again, r is the distance from the current element to where we're measuring the field, and phi is the angle between the direction of the current element and the line pointing to that measurement point.

So the same idea of the angle mattering and the field strength weakening with distance.

Exactly.

And just like with the single charge, there's also a vector form of the Biot -Savart law to tell us the direction of this tiny magnetic field.

dB equals O over 4 times iok cross product r hat over r squared.

Oh boy, more cross products.

Don't worry, the right hand rule is here to help us again.

Point your right thumb in the direction of the current element, little edgel, and your fingers towards the point where you want to find the field.

Okay, thumb for current direction, fingers towards the point.

And just like before, your curled fingers will then show you the direction of that tiny magnetic field Perfect.

So we've got a tiny piece of current making a tiny magnetic field that curls around it, but a whole wire is made of tons of these tiny pieces.

That's right.

So to get the total magnetic field created by the whole wire, no matter what shape it has, we need to add up the contributions from all those tiny current elements.

I'm sensing another integral coming on.

You got it.

We use an integral to sum up all those dB contributions.

The integral from the Biot -Savart law gives us total magnetic field B at any point around the wire.

Sounds like it can get complicated depending on the shape of the wire.

It can, but luckily for some common shapes like a long straight wire, the math works out quite nicely.

Okay, so let's take that example.

What's the magnetic field around a long straight wire carrying a current?

Well, if we do all the integral magic from the Biot -Savart law, we get a very elegant result.

The magnitude of the magnetic field at a distance r from the wire is B equals u over 2 par.

Much simpler than I expected.

So the field strength is directly proportional to the current and inversely proportional to the distance.

Makes sense.

It does.

And the direction of the field is really cool too.

It forms concentric circles around the wire.

So the field loops around the wire like a bunch of invisible rings.

Exactly.

And to figure out which way those loops go, we use another version of our trusty right -hand rule.

Point your right thumb in the direction of the current in the wire.

Thumb for current direction.

And your curled fingers will show you the direction of those magnetic field lines looping around the wire.

Awesome.

So we've got these magnetic fields swirling around wires.

What happens when you have more than one wire near each other, each with their own swirling fields?

Ah, now that's where things get even more interesting.

It turns out those magnetic fields interact with each other.

So like the fields from different wires can affect each other.

They do.

And it leads to some pretty remarkable effects.

Imagine two long straight wires running parallel to each other, both carrying currents.

Okay, I'm picturing it.

Each wire has its own magnetic field circling around it.

Right.

Now if the currents in those wires are flowing in the same direction, their magnetic fields will actually cause them to attract each other.

Like the wires themselves will move closer together.

Yep.

They'll experience a force pulling them together.

But if the currents are flowing in opposite directions, then the wires will repel each other.

Whoa.

So the direction of the current determines whether they attract or repel.

That's pretty wild.

It is.

And we can even calculate the strength of that force between the wires.

It depends on the currents in each wire, the distance between them, and of course our old friend, me.

What's the equation for that?

It's a bit of a fall.

The force per unit length, FLL, between the wires is equal to UO times II over 2R, where I and I are the currents in the two wires and R is the distance between them.

So stronger currents or closer wires mean a stronger force.

Makes sense.

It does.

And here's a really cool fact.

That equation for the force between wires is actually how we define the ampere, the unit of electric current, in the SI system of units.

No way.

So the very definition of an ampere is based on this magnetic interaction between wires.

That's mind -blowing.

It is.

It shows how deeply interconnected electricity and magnetism truly are.

So we've talked about straight wires.

What if we bend a wire into a loop?

What happens to the magnetic field then?

Ah, when you bend the wire into a loop, the magnetic field changes quite a bit.

Instead of those nice, even circles, it becomes a bit more complex.

More complex.

How so?

Well, the field still curls around the wire, but it starts to look more like a dipole field, similar to what you'd see around a bar magnet.

Okay, so like a mini bar magnet with a north and south pole.

Exactly.

And the strength of the field at different points around the loop can be calculated using an equation derived from the biots of art law.

It depends on the current, the radius of the loop, and the distance and angle from the loop.

Can you give us an example of that equation?

Sure.

For a circular loop with radius A carrying a current I, the magnetic field at a point X along the axis of the loop is given by Billy X equals E S divided by two times the quantity X force A plus A raised to the power of 32.

That definitely looks a bit more involved than the straight wire equation.

It is, but it's a powerful equation that lets us calculate the field at any point along the axis of the loop.

And it gets even more interesting if you have multiple loops stacked together, like in a coil.

Like the solenoid.

Exactly.

If you have N identical loops forming a coil, the magnetic field at the center of the coil is just N times the field of a single loop.

So more loops mean a stronger field at the center.

Precisely.

And the direction of the field inside the coil follows a familiar pattern.

It's aligned along the axis of the coil.

And I bet there's a right hand rule for this too.

You bet there is.

Curl the fingers of your right hand in the direction of the current flowing in the loop.

Okay, fingers curling with the current.

And your thumb will point in the direction of the field inside the loop.

Thumb for field direction.

Got it.

Now you mentioned something earlier called the magnetic dipole moment.

What exactly is that?

It's a really important concept.

It's a vector quantity that basically describes how strong a magnetic dipole is and how it's oriented.

So how magnetic the loop is and which way it's pointing.

Exactly.

For a single loop with current I and area A, the magnetic dipole moment M is simply A times A.

And if you have N loops, it's just N times IA.

Okay, so more current, larger area, or more loops all mean a bigger magnetic dipole moment.

Precisely.

And it's measured in amperimeters squared.

It's this magnetic dipole moment that really governs how a current loop will interact with other magnetic fields, including the Earth's magnetic field.

Now we've been using the Biot -Savart law to calculate these fields, which get pretty complex.

Is there an easier way, a shortcut for certain situations?

There is, especially when you're dealing with systems that have a lot of symmetry.

That's where Ampere's law comes in.

Ampere's law.

Another law of magnetism.

It's a powerful one.

It basically says that the line integral of the magnetic field around a closed loop is directly proportional to the total current passing through that loop.

Okay, line integral.

That sounds a bit scary.

Can you break that down?

Sure.

Imagine drawing a closed loop, any shape you like.

We call this an imperial loop.

Okay, a loop in space.

Now imagine taking tiny little steps along that loop, and at each step you look at the magnetic field and how much of it is pointing along your path.

So like the component of the field that's aligned with our little step.

Exactly.

Ampere's law says that if you add up all those little contributions of the magnetic field along your entire loop,

that sum will be equal to i times the total current passing through any surface bounded by your loop.

So the magnetic field around the loop is connected to the current passing through it.

Precisely.

It's a very elegant relationship.

And just like with the Biot -Savart law, we have a right -hand rule to help us with the directions.

If your fingers curl in the direction you're going around the loop, your thumb points in the direction of the positive current.

Okay, fingers for loop, direction, thumb for current direction.

Got it.

Now you mentioned that Ampere's law is particularly useful for systems with symmetry.

Can you give us an example?

Absolutely.

Let's go back to that long, straight wire carrying a current.

That's a very symmetrical situation.

Right.

The field just loops around the wire evenly.

Exactly.

So if we choose a circular Amperian loop centered on the wire, the magnetic field will have the same magnitude at every point on the loop, and it will be tangent to the loop at every point.

So the field and our little steps are always aligned.

That's right.

And that makes the math super easy.

Ampere's law tells us that the magnetic field times the circumference of the loop is equal to A i's times the current in the wire.

And from that, we can solve for the magnetic field.

Yep.

And you get the same equation we got before using the Biot -Savart law.

B equals E i over 2 bar, but with much less effort.

Nice shortcut.

Can you give us another example of how Ampere's law simplifies things?

Sure.

Let's look at a solenoid, that tightly wound coil of wire we talked about earlier.

Solenoids, they create those nice, uniform magnetic fields inside, right?

They do.

And Ampere's law helps us see why.

Imagine drawing a rectangular Amperian loop that goes through the solenoid with one long side inside the solenoid and one long side outside.

Okay, a loop that cuts through the coil.

Right.

The key is that the magnetic field outside a solenoid is very weak, practically zero.

So when we apply Ampere's law, we only need to consider the contribution from the part of the loop that's inside the solenoid.

And the field inside is nice and uniform.

Exactly.

So Ampere's law gives us a simple equation relating the field strength inside to the current and the number of turns per unit length of the solenoid.

B equals Uran, where N is the number of turns per unit length.

So the field inside a solenoid is directly proportional to the current and how tightly the wire is wound.

Makes sense.

It does.

And another cool example is a toroid, which is like a donut -shaped solenoid.

The donut -shaped coil of wire.

Exactly.

And because of its shape, the magnetic field is completely contained inside the toroid.

It's zero outside.

So all the magnetic field is trapped within the donut hole.

Precisely.

And Ampere's law lets us easily calculate the field strength inside the toroid.

It's B equals Euro over two far, where N is the total number of turns and R is the distance from the center of the donut hole.

So the field gets weaker as you move further from the center of the Exactly.

Ampere's law is really powerful for analyzing these symmetric situations.

It makes the calculations much more manageable.

So we've seen how moving charges create magnetic fields,

how currents and wires generate these fields, and how Ampere's law helps us calculate them in symmetrical cases.

But what about the materials themselves?

Can they be magnetic?

They absolutely can.

And it all comes down to what's happening at the atomic level.

Okay.

So back to the tiny world of atoms.

Exactly.

We know that electrons are constantly in motion within atoms.

They're orbiting the nucleus and they're also spinning on their own axis.

So lots of tiny moving charges within each atom.

That's right.

And since moving charges create magnetic fields, each electron in an atom acts like a tiny little magnet.

Wow.

So every atom has the potential to be magnetic.

In a sense, yes.

And the strength and orientation of that tiny atomic magnet is described by its magnetic dipole moment.

Okay.

So each atom has its own magnetic dipole moment, like a tiny little compass needle.

Exactly.

But here's the thing.

In most materials, these atomic dipole moments are all randomly oriented, pointing every which way.

So they all cancel each other out.

Pretty much.

The magnetic fields from all those randomly oriented atomic magnets just average out.

So the material as a whole doesn't show any overall magnetism.

But what if they're not randomly oriented?

What if you can get them to line up?

Ah, now you're talking.

That's when things get really interesting.

That's where we get into different types of magnetic materials.

Paramagnetic, diamagnetic and ferromagnetic.

Okay.

So the way those atomic magnets align or don't align determines the magnetic behavior of the material.

Exactly.

Let's start with paramagnetic materials.

Paramagnetic, those are the ones that are weakly attracted to magnets, right?

That's right.

In paramagnetic materials, the atoms do have those intrinsic magnetic dipole moments, but they're usually randomly oriented, so no overall magnetism.

But something changes when you bring a magnet near them.

Exactly.

When you apply an external magnetic field, those atomic magnets tend to align themselves with the field.

So they try to line up with the external field.

Yep.

And that alignment adds to the overall magnetic field, making it a bit stronger inside the material.

But it's a weak effect, right?

It is.

Paramagnetic materials have a small positive

susceptibility, which is a measure of how easily they become magnetized.

Okay.

So they're slightly magnetic, but not super strong.

What about diamagnetic materials?

How are they different?

Diamagnetic materials are kind of weird.

Their atoms don't have those permanent magnetic dipole moments.

So no tiny magnets within the atoms.

Not in the usual sense.

But when you apply an external magnetic field, it actually induces tiny temporary magnetic dipole moments in the atoms.

Induced.

Meaning they're created by the external field.

Exactly.

And here's the kicker.

These induced moments are aligned in the opposite direction to the external field.

So they actually oppose the external field.

They do.

It's like they're trying to push back against the magnetic field.

So diamagnetic materials are weakly repelled by magnets.

That's right.

It's a very weak effect, but it's there.

And unlike paramagnetism, diamagnetism is pretty much independent of temperature.

Okay, so those are the weird ones.

Now finally, let's talk about ferromagnetic materials.

These are the ones we usually think of as magnetic, right?

Exactly.

Ferromagnetic materials are the rock stars of magnetism.

Iron, nickel, cobalt, those are the classic examples.

And they can become permanent magnets, like the ones on our fridge.

Exactly.

So what makes ferromagnetic materials so special?

Well, the interaction between neighboring atoms is much stronger in these materials.

Right, or how?

Their atomic magnets strongly influence each other, causing them to spontaneously align even without an external magnetic field.

So they line up on their own, forming little domains of aligned magnets.

Exactly.

These are called magnetic domains, and each domain has a bunch of atomic magnets all pointing in the same direction.

So it's like the material is already pre -magnetized in these little regions.

So that's a good way to think about it.

Now when you apply an external magnetic field, those domains that are already aligned with the field grow bigger.

Like they're absorbing the other domains.

Kind of.

And the domains themselves can also rotate to align with the field.

So you get this huge increase in the overall magnetization of the material.

So that's why ferromagnetic materials are so strongly attracted to magnets.

Precisely.

And they also have this cool property called hysteresis.

Hysteresis, that's a new one.

What is that?

It means that the magnetization of the material depends not only on the current magnetic field, but also on its past history, on the fields it's experienced before.

So it remembers its magnetic past.

In a way, yes.

And that's why ferromagnetic materials can become permanent magnets.

Even after you remove the external field, some of those domains stay aligned, keeping the material magnetized.

That's pretty amazing.

Now when we're doing calculations with magnetic fields inside these materials, do our equations change at all?

Yes, they do.

Instead of using the permeability of free space, we use the permeability of the materials have different permeabilities.

Exactly.

The permeability basically tells you how easily a magnetic field can be established within the material.

And for ferromagnetic materials, that permeability can be much larger than me off, right?

That's right.

It can be hundreds or even thousands of times larger, which is why they can become such strong magnets.

Okay.

So we've covered a ton of ground today from tiny moving charges to powerful magnets.

It's been quite a journey through the world of magnetism.

It has.

We started with the fundamental building blocks, those moving charges creating magnetic fields, and then we saw how currents in wires generate these fields.

And we learned about Ampere's law, which helps us analyze those fields in symmetrical situations.

Then we went deeper into the materials themselves, seeing how the behavior of those tiny atomic magnets determines whether a material is paramagnetic, diamagnetic, or ferromagnetic.

And how ferromagnetic materials with their magnetic domains are the key to creating permanent magnets.

Exactly.

We've covered all the major points from the chapter on sources of magnetic fields.

Awesome.

So as we wrap up, what's the key takeaway for our listeners?

Why is understanding all of this important?

Well, I think the key takeaway is that magnetism is everywhere.

From the tiny world of atoms to the vast magnetic field of the Earth, it's a fundamental force that shapes our world.

Right.

And it's the foundation for so much of our technology, from electric motors to MRI machines.

Exactly.

So next time you see a magnet or use a device that relies on magnetism, remember that it all goes back to these fundamental principles we've talked about today.

Now for our final thought, let's leave our listeners with something to ponder.

We've been talking about classical magnetism, but what happens when you go even smaller, down to the quantum world?

That's a great question.

In the realm of quantum mechanics, things can get pretty strange.

The rules that govern magnetism at atomic and subatomic level can be quite different from the classical laws we've discussed.

So there's a whole other level of complexity to explore there.

There is.

The interaction between moving charges and magnetic fields becomes even more intricate and fascinating in the quantum world.

It's a reminder that there's always more to learn, more to discover, about the universe around us.

Absolutely.

And who knows what incredible technologies await us as we unlock the secrets of quantum magnetism.

Well, that's a great note to end on.

Thanks for joining us for this deep dive into the sources of magnetic fields.

It's been my pleasure.

Until next time, keep exploring the fascinating world of science.

And keep those questions coming.

We'll be back with another deep dive soon.