Chapter 27: Magnetic Field and Magnetic Forces

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Have you ever thought about how much we rely on magnets?

I mean, they're everywhere.

How would we even listen to music without speakers?

Exactly.

Or use a compass or a motor or even store data on a hard drive.

It's wild.

Yeah.

For this deep dive, let's explore the core principles of magnetic fields and forces.

We've got some great resources that really unpack this whole invisible world.

Yeah.

The goal today is to understand how magnetism is actually generated and how those magnetic forces work.

Right.

How they interact with moving charges and currents and all that.

Okay.

So, first things first.

Where does magnetism even come from?

Well, the most fundamental thing to remember is that magnetism is all about moving electric charges.

So, it's like if you have electrons zipping around in a wire?

Exactly.

That movement, that flow of charge, it actually creates a magnetic field around the wire.

And then that field can push or pull on other moving charges or currents, right?

That's it.

It's like a two -step process.

First, the moving charge or the current creates the field.

And then that field reaches out and interacts with other moving charges.

And it's important to remember that this is different from static electricity, right?

Right.

That static cling you get from rubbing a balloon on your hair, that's from charges that are just sitting there, not moving.

So magnetism is all about motion, right?

Absolutely.

And when we talk about magnets, we usually picture those classic north and south poles.

Yeah.

Opposites attract, like poles repel.

Right.

Just like with positive and negative electric charges.

But here's a big difference.

Magnetic poles always, always come in pairs.

Like you can't have a north pole without a south pole.

Exactly.

No one has ever found a magnetic monopole.

Imagine trying to find a coin that only has heads.

That's a really good way to put it.

It just doesn't exist in the world of magnets.

And this actually gets into how permanent magnets, like the ones on our fridge, work.

Oh yeah.

How does a chunk of iron just become a magnet?

Well, deep down at the atomic level, it comes from the coordinated motion of electrons within the iron atoms.

So even in a solid chunk of metal, it's still the movement of these tiny charged particles that creates the magnetism.

It's pretty amazing when you think about it.

Okay.

So let's talk about the magnetic field itself.

It's not just some vague force field, right?

It's a measurable thing.

Oh, absolutely.

The magnetic field, which we usually represent with a big B, is what's called a vector field.

Meaning?

It means at any point in space, the magnetic field has both a strength or intensity and a direction.

Okay.

I'm following.

So how do we figure out which way the magnetic field is pointing?

Well, we use a convention.

The direction of the magnetic field at a particular point is defined as the direction that the north pole of a tiny compass needle would point, if you put it there.

Oh, like those pictures you see with iron filings around a magnet.

They all line up along the field lines.

Exactly.

And those lines give you a visual representation of the shape and strength of the field.

For a bar magnet, the lines are usually drawn coming out of the north pole and looping back into the south pole.

So a continuous loop, basically.

Okay.

Now what about the actual force that a magnetic field exerts?

Ah, yes.

Imagine a charged particle, let's say it has a charge of Q, and is moving with a velocity V through a magnetic field B.

It's going to feel a force from that field.

And the formula for that force is given by a vector product.

Have you ever worked with those?

I think I vaguely remember them from physics class.

Well, the key takeaway here is that the magnetic force is always perpendicular to both the direction the particle is moving and the direction of the magnetic field itself.

So it's kind of like, if you imagine the particle's motion and the magnetic field forming a flat plane,

the force pops out at a right angle to that plane.

Precisely.

It's a three -dimensional relationship.

And how do we figure out how strong that force is?

There's a formula for that, too.

The magnitude of the force, so how strong it is, is given by FQVB sin.

Okay, hold on.

What's that fin to the formula?

Right.

That is the angle between the particle's velocity and the magnetic field.

So if the particle's moving perfectly along the field lines or exactly opposite to them, the angle is zero, or 180 degrees, and the sine of that is zero, which means no force.

Exactly.

The magnetic force is strongest when the particle's velocity is perpendicular to the field.

That's when it's 90 degrees, and the sine of 90 degrees is one.

Okay, that makes sense.

But how do we know which way the force actually pushes or pulls the particle in three -dimensional space?

We have a handy tool for that called the right -hand rule.

Oh, right.

I remember vaguely hearing about that.

It's super helpful.

Basically, you point the fingers of your right hand in the direction the positive charge is moving.

Then you curl your fingers towards the direction of the magnetic field,

and your thumb will then point in the direction of the magnetic force on that positive charge.

And if it's a negative charge?

Then the force is in the opposite direction.

Okay, so we can figure out the direction and the strength of the force, but how do we actually measure the strength of the magnetic field itself?

Well, the SI unit for magnetic field strength is the tesla, abbreviated as T.

A tesla, okay.

And what does that actually represent?

One tesla is the strength of a magnetic field that would exert one newton of force on a one ampere current flowing through a one meter wire perpendicular to the field.

Okay, so it ties back to that force we were just talking about.

Exactly.

It's all connected.

Now, let's dive a bit deeper into those magnetic field lines.

We know they help us visualize the field, but they're different from electric field lines, right?

Right.

And it's a key difference.

Magnetic field lines always form closed loops.

They don't have a starting point or an ending point.

And that's because there are no magnetic monopoles, right?

Precisely.

You can't have a field line just ending abruptly in space because there's no isolated north or south pole for it to start or end on.

So it's a continuous flow.

That's it.

Yeah.

And to describe how much of this magnetic field is actually passing through a given surface, we use the concept of magnetic flux.

Magnetic flux, okay.

Yeah.

It's often represented by the Greek letter phi with a subscript B.

Mathematically, it's the integral of the magnetic field component perpendicular to the surface over the entire surface area.

But for a simple case, let's say you have a uniform magnetic field passing through a flat surface.

The flux is just the magnetic field strength multiplied by the area of the surface multiplied by the cosine of the angle between the field and the normal to the surface.

Hold on.

What's the normal to the surface?

Oh, right.

Good question.

The normal is just a line that's perpendicular to the surface at that point.

Okay.

So we're basically taking into account how much of the magnetic field is directly piercing through the surface.

Exactly.

And the unit for magnetic flux is the Weber, abbreviated as one tail.

One Weber is equivalent to one Tesla times one square meter.

Okay, got it.

Now this idea of closed loops and no -monopoles, is this like a super important rule in magnetism?

It is.

It's one of the fundamental laws of electromagnetism, Gauss's law for magnetism.

Which says?

It states that the total magnetic flux through any closed surface is always zero.

It means that however many magnetic field lines enter a closed surface,

the same number must also exit.

There can't be any net creation or destruction of magnetic field lines within that closed surface.

So it kind of reinforces this idea that isolated magnetic poles don't exist.

Exactly.

It's a really big deal in how we understand magnetic fields and how they're generated.

Okay.

So we've talked about fields and flux.

Now, what happens when we actually have charged particles moving around in these magnetic fields?

Ah, good question.

Remember that the magnetic force is always perpendicular to the velocity of the charge, right?

Right.

Well that means the force can only change the direction of the particle's motion.

It can't make it speed up or slow down.

Because the force isn't pushing it forward or backward, it's pushing it sideways.

Exactly.

So the kinetic energy of the particle, which is related to its speed, stays the same.

So what kind of paths can these particles take in a magnetic cloud?

Well, it depends on the orientation of the velocity in the field.

If a charged particle enters a uniform magnetic field and its velocity is exactly perpendicular to the field lines, it's going to move in a circle.

A perfect circle.

Yep.

The magnetic force is always acting towards the center of the circle, like the tension in a string when you swing a ball around.

Okay.

And is there like an equation for the radius of this circle?

There is.

The radius is given by r equals mvqb.

So it depends on the mass m of the particle, its charge q, its velocity v, and the strength of the magnetic field b.

Okay.

That makes sense.

So what if the particle's velocity isn't exactly perpendicular to the field?

If there's a component of the velocity that's parallel to the field lines, then that component isn't affected by the magnetic force at all.

It just keeps moving along the field lines at a constant speed.

So you get a combination of circular motion and straight line motion?

Exactly.

The particle ends up tracing out a helix.

It's like a spiral or a corkscrew.

Yeah.

And these types of motion have led to some really interesting applications.

One example is a device called a velocity selector.

What's that?

Imagine setting up a region with both an electric field and a magnetic field.

And they're arranged so that they're perpendicular to each other and to the path you want the particles to take.

If you adjust the strength of the electric and magnetic fields just right,

only particles with the specific velocity will pass through undeflected.

How does that work?

Well, for particles with that specific velocity, the electric force and the magnetic force end up being equal in magnitude, but opposite in direction.

So they cancel each other out.

So any particle that's going too slow or too fast will get deflected off course?

That's right.

And this velocity selector is super useful in all sorts of things, from old CRT TVs to modern scientific instruments.

Very cool.

And another really cool application is the mass spectrometer.

Ah, I've heard of those, but I don't really know what they do.

Well, a mass spectrometer typically uses a velocity selector as a first step to make sure all the ions entering the instrument have the same velocity.

Okay, and then?

Then those ions enter another magnetic field where they get deflected into semicircular paths.

And the radius of those paths depends on the ion's mass to charge ratio.

So by measuring how much they curve, you can figure out how massive they are.

Exactly.

And even separate isotopes of the same element, which has slightly different masses.

Wow, that's incredibly precise.

It's a powerful tool used in chemistry,

biology, environmental science, forensics, you name it.

That's amazing.

And it's all based on the simple principles of how charged particles move in magnetic fields.

It all comes back to the basics.

Okay, so far we've been talking about forces on individual moving charges.

But what happens when you have a whole bunch of charges moving together, like in an electric current?

Well, a current carrying wire in a magnetic field will also experience a force.

After all, a current is just a stream of moving charges, and each of those charges is subject to the magnetic force.

Right, so the whole wire feels it.

Exactly.

And if it's a straight wire segment of length carrying a current I in a uniform magnetic field B,

the force on the wire is given by another vector product, F equals I L A B.

Okay, and here the L represents the length and direction of the wire.

You got it.

The vector L points along the wire in the direction of the current flow.

And just like before, the force is perpendicular to both the current and the magnetic field.

That's right.

Imagine a boat trying to move straight across a river with a strong current.

The current will push the boat sideways.

Okay, I get the analogy.

Now what if the wire isn't perfectly straight?

Like what if it's all curvy?

Good question.

For those cases, you can break the wire down into tiny little straight segments, and each segment will experience a tiny force.

Then you just add up all those tiny forces, which mathematically means doing an integral.

Sounds a little complicated.

It can be, but the basic idea is the same.

Moving charges in a magnetic field feel a force, whether it's a single charge or a whole bunch of them in a current.

And we actually use this principle all the time, right?

Oh yeah.

Think about loudspeakers.

They have a coil of wire called a voice coil that carries a current.

And that coil sits in a magnetic field.

And when the current changes, the magnetic force on the coil changes, making the speaker cone vibrate.

And that creates the sound waves.

Exactly.

So all that music we listen to is thanks to the magnetic force on a current carrying wire.

That's pretty cool.

Okay, so what happens when we take that current carrying wire and make it into a loop?

Now we're getting to the heart of how motors work.

For real?

Yeah.

When you put a current loop in a uniform magnetic field, it doesn't necessarily move in a straight line.

Wait, really?

Why not?

Because the forces on opposite sides of the loop can end up canceling each other out.

But it will experience a torque.

A torque meaning?

A twisting force that tries to make the loop rotate.

Oh, I see.

And why does that happen?

Well, to really understand that, we need to introduce the concept of the magnetic dipole moment.

Magnetic dipole moment, huh?

Yeah, it sounds a bit intimidating, but it's basically a measure of the loop's overall magnetic strength and orientation.

So a bigger loop with more current has a stronger magnetic moment.

Exactly.

It's represented by a vector, usually a capital M.

Okay, I'm following.

So how do we calculate it?

For a single lock of wire, the magnetic dipole moment is just the current in the loop, I, times the area enclosed by the loop, A.

So M equals IA.

And if you have like a coil with many turns of wire?

Then you just multiply that by the number of turns.

Okay, that makes sense.

Now, what does this magnetic moment have to do with the torque on the loop?

Well, the torque on the loop is given by another vector product.

This time, it's the magnetic moment M crossed with the magnetic field B.

So A equals MAB.

And the strength of that torque depends on the angle between the magnetic moment and the magnetic field, right?

You got it.

The torque is strongest when they're perpendicular and zero when they're aligned or anti -aligned.

So the torque is always trying to twist the loop to line up its magnetic moment with the external field.

Exactly.

It's a lot like a compass needle trying to align itself with the Earth's magnetic field.

Oh yeah, that makes sense.

And just like the compass needle has potential energy, depending on its orientation relative to the Earth's field,

a current loop also has potential energy in an external magnetic field.

So the loop's most stable when its magnetic moment is perfectly aligned with the field.

That's right.

It has the lowest potential energy in that configuration.

And the highest potential energy when they're anti -aligned.

You got it.

And this whole idea of a current loop behaving like a magnet actually helps explain how permanent magnets can attract things like iron.

Oh yeah.

How does that work again?

Well, the magnetic field of a permanent magnet can influence the tiny magnetic moments of the atoms within the iron.

It can align them, creating a net magnetic moment in the iron, and then that induced moment gets attracted to the permanent magnet's field.

So it's kind of like the permanent magnet creates a temporary magnet in the iron.

Precisely.

And that's what we see is the attraction between them.

That's super cool.

Okay, so now we finally get to motors, right?

Yeah.

It all builds up to this.

This principle of a torque acting on a current loop is the very foundation of a direct current or DC motor.

Oh, cool.

So how does that actually work?

The basic idea is you have a coil of wire called the rotor that sits in a magnetic field created by either permanent magnets or electromagnets.

This whole assembly is called the stator.

When current flows through the coil, it experiences a torque that makes it rotate.

Okay, that makes sense so far, but how do you keep it rotating continuously?

Wouldn't it just like flip back and forth?

That's where the magic of the commutator and brushes comes in.

The what and the what?

The commutator and brushes.

They're the key to continuous rotation.

The commutator is a split ring that's connected to the coil, and the brushes are stationary contacts that rub against the commutator as it spins.

So as the coil rotates, the commutator segments switch which brush they're in contact with, and this switching actually reverses the direction of the current in the coil every half turn.

Oh.

Yeah.

So it's like constantly flipping the direction of the current so that the torque always keeps pushing the coil in the same direction.

Sicely.

Brilliant.

Super clever.

It is.

And as the motor spins, something else really interesting happens.

A back electromotive force, or back F,

is induced in the rotating coil.

Back EMF?

What's that?

It's a voltage that's generated in the coil as it rotates through the magnetic field, and it actually opposes the applied voltage that's driving the motor.

Oh, interesting.

So it's like a kind of resistance.

Exactly.

It's like a built -in braking system.

The faster the motor spins, the greater the back EMF, and the less current it draws from the power source.

So it kind of regulates itself.

In a way, yes.

Yeah.

The back EMF helps balance things out.

Okay, that's pretty wild.

Now, our deep dive also mentions this thing called the Hall Effect.

What's that all about?

Ah, the Hall Effect.

It's a really interesting phenomenon.

Imagine you have a conductor with current flowing through it, and you put it in a magnetic field that's perpendicular to the current.

Well, the charge carriers in the conductor, whether they're electrons or something else, will experience a magnetic force, and that force will deflect them to one side of the conductor.

So they all kind of bunch up on one side?

Right.

You end up with a buildup of charge on one side and a lack of charge on the other.

And this creates a voltage across the conductor.

We call this the Hall Voltage.

Okay, and what's so special about this Hall Voltage?

Well, by measuring it, you can actually figure out a lot about the material the conductor's made of.

Really?

Yeah.

You can determine the sign of the charge carriers, meaning whether they're positive or negative, and their concentration, or how many there are per unit volume.

That's pretty amazing.

It's a really powerful technique for characterizing materials.

Okay, before we wrap up, our source material also includes a glossary of terms, which I always find super helpful.

It's great for making sure we really understand all the lingo, from basic things like magnetic field and magnetic force to more advanced stuff like dipole moments and back EMF.

Absolutely.

It's a great resource to brush up on the vocabulary of magnetism.

So let's sum up our deep dive.

We started with the fundamental idea that magnetism comes from the motion of electric charges, which create magnetic fields.

Right.

And these fields then exert forces on other moving charges and currents.

We talked about how magnetic poles always come in pairs.

And no monopoles allowed.

We discussed how magnetic field lines form closed loops, and how we use magnetic flux to describe the amount of field passing through a surface.

And we explored all the amazing ways charged particles move in magnetic fields, like circles, helices, and all the applications that come from that.

Like velocity selectors and mass spectrometers.

We also covered how current carrying wires experience forces and torques in magnetic fields.

Which is how motors and speakers work.

We learned about magnetic dipole moments and how they act like tiny bar magnets.

And finally, we even touched on the Hall effect and how it helps us understand materials.

It's incredible how much is packed into this concept of magnetism.

It's everywhere.

From the tiniest atoms to the vastness of space.

And it's absolutely essential for so much of our technology.

Absolutely.

And you know, we just scratched the surface here.

We didn't even talk about the Earth's magnetic field, which protects us from harmful radiation from space.

Or how we use magnetism to store data on hard drives and other devices.

There's so much more to explore.

And one thing that always blows my mind is the connection between electricity and magnetism.

We haven't even touched on electromagnetic induction.

Where a changing magnetic field can create an electric field and vice versa.

It's the basis for power generation and wireless communication and so much more.

Exactly.

It's mind boggling.

Maybe we can do another deep dive on that sometime.

But for now, I hope you enjoyed this journey into the world of magnetic fields and forces.

It's a truly fascinating subject.

It is.

Thanks for joining us on this deep dive.

Thanks everyone.