Chapter 17: The Laws of Induction & Electromagnetic Force

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All right, welcome back.

Today we're going to really dig into the laws of induction.

We're looking at a classic chapter trying to get past just the formulas, you know.

Yeah, absolutely.

We want to get to the core physical ideas because when you mention electromagnetic conduction, everyone jumps to Faraday's law, right?

EMF equals minus di phi dt.

Exactly.

The flux rule.

But the crucial insight here is that this single equation actually covers two totally different ways things can happen physically.

It's not one phenomenon.

And understanding that difference is, well, it seems like it's key to really getting electromagnetism.

The goal isn't just the flux rule.

It's seeing how it all ties back to the fundamental force law.

The Lorentz force.

$5 equals Q plus V times B.

That's the bedrock.

That equation has to explain everything, even the tricky bits we're about to get into.

Okay.

So let's start with that famous flux rule.

Math go, math to kitchen.

Simple idea.

Change the magnetic flux through a loop and you get a voltage and EMF around it.

Right.

And it works beautifully in like 99 % of cases you encounter in basic circuits.

The math holds up.

But the physics underneath, that's where the nuance is.

That's it.

Feynman really forces us to see the two distinct causes baked into that one rule.

So case one.

Case one.

The wire, the conductor is moving, but the magnetic field B is staying put.

It's constant.

Okay.

So the charges inside that wire are moving with velocity $5 through the field by dollars.

They feel force, the by dollars times B by part of the Lorentz force.

That pushes them along the wire.

That's your EMF.

So mechanical force really motion through a field.

Precisely.

Now case two, the wire isn't moving at all.

It's stationary, but the magnetic field itself is changing in time.

Okay.

So DBDT isn't zero.

Right.

And Maxwell's equations tell us that a changing B field must create an electric field, an E field, an induced E field.

Exactly.

And that E field is what pushes the charges around the stationary wire.

No by dollar times B involved here.

It's purely the dollar part of the Lorentz force.

Two totally different physical mechanisms, both described by the same flux rule.

Yeah.

And that's why just blindly applying the flux rule can sometimes, you know, make you miss what's actually going on, which leads to the so -called paradoxes.

Right.

The exceptions that prove the rule,

or rather show the limitations of the simple rule.

Let's talk about the rotating disc, the homopolar generator idea.

Okay.

Yeah.

So imagine a flat conducting disc, like a copper plate spinning around its center, and it's sitting in a uniform magnetic field pointing straight up perpendicular to the disc.

Okay.

Spinning disc, steady B field.

We hook up wires, say one to the axle and one to the outer rim to make a circuit.

Now, if you try to calculate the magnetic flux, phi s, through that circuit, well, the B field is constant.

The area isn't really changing in a simple way that makes the flux change.

Depending on how you draw the loop, the flux might even look like it's zero or constant.

So the flux rule, DVFVU, would predict zero EMF.

It seems like it should, but you definitely measure a voltage.

There's a real steady EMF between the center and the rim.

So it's purely the bi -dollar times B -vale force again.

The charges in the disc are moving outwards, or rather circularly, with velocity dollars through the fixed B field.

That force pushes charge radially, creating the potential difference.

It's entirely about the motion of the conductor.

The flux rule, applied naively, kind of misses the point there.

Okay.

That one's cleared up by bi -dollar times B -gool.

What about the other example?

The rocking plates.

That one seems even weirder.

It is subtle.

Imagine two parallel metal plates hinged at one end and they rock back and forth, opening and closing like a clamshell within a uniform magnetic field.

Okay.

So the area between them is changing significantly.

Right.

And you connect a galvanometer, a voltmeter, with wires that run outside the region where the plates are moving.

So the meter and its wires are stationary.

Now the area is changing.

B is constant.

So the flux flee through the loop formed by the plates and the meter connection is definitely changing a lot.

So the flux rule, natal martyr, predicts a big oscillating EMF should be measured by the galvanometer.

But you measure nothing.

Zero EMF if those connections are truly stationary and outside the moving region.

How can the flux change so much but the EMF around that path be zero?

Because the EMF fundamentally is the line integral of the total force per unit charge around the loop.

That's L E plus V times B C I I S A C S.

The flux rule is a shortcut for this integral.

In the rocking plates case, yes, there is an induced E field in the region where B is changing or where the boundary moves.

But if your voltmeter's path stays outside that region, the integral of that specific E field along that specific path is zero.

The va dollar times B part is also zero along the stationary wires.

So no net EMF around that chosen path.

So the flux rule gives the total change but doesn't tell you where the induced field is.

You need the full Lorentz law.

Exactly.

The physics is always local.

Va dollar equals Q at plus V times B tells you the force on each charge right then and there.

The flakes rule is convenient but it's not the fundamental physical cause in the same way.

It averages things out.

Okay, so if we want to really isolate the effect of a changing B field creating an E field without any moving wires, val dollars times B confusing things, where do we look?

Well, that brings us to a fantastic piece of technology.

The Betatron.

It's basically designed to exploit exactly that phenomenon.

It's a particle accelerator.

It speeds up electrons, doesn't it?

It does.

It accelerates electrons to very high energies keeping them in a circular path and the key is it uses a magnetic field that's specifically designed to change in time and it's symmetric around the center axis.

So how does the changing B field accelerate the electrons?

Because a changing magnetic flux through the center of the electron's orbit induces an electric field and this E field points tangentially exactly along the direction the electron is moving.

So it constantly pushes the electron making it go faster and faster.

Okay, that's the acceleration part but a magnetic field is also needed to bend the electrons into a circle, right?

The centripetal force.

Precisely.

So the magnetic field has to do two jobs simultaneously.

The changing flux in the middle creates the accelerating E field but the magnetic field at the orbit radius itself has to provide the Jule V times B dollar force needed to keep the electron on that nice circular track.

That sounds tricky.

As the electron speeds up, doesn't it need a stronger B field to keep it in the same circle?

But the changing field is also accelerating it.

How did that balance?

Ah, that's the genius of it.

There's a critical condition.

For the electron to stay in an orbit of a constant radius while it's being accelerated, there's a very specific relationship required between the field at the orbit and the average field inside the orbit.

What's the relationship?

The average magnetic field inside the orbit must be exactly twice the magnetic field right at the

The electrons will spiral either inwards or outwards and hit the walls of the vacuum chamber.

It's a beautiful piece of applied physics relying entirely on harnessing that induced electric field from a changing magnetic field.

Amazing.

Okay, so that's high energy physics.

What about something more every day, like power generation?

Well, the standard alternating current generator uses the other aspect of induction, mostly the Vi dollar times B but in a rotational way.

How does that work?

You have a coil of wire spinning.

Yeah, typically you have a coil of wire rotating within a fixed uniform magnetic field.

As the coil spins, the angle between the coil's surface area vector and the magnetic field lines changes continuously.

And that changing angle means the flux through the coil changes.

Right.

The magnetic flux passing through the coil goes up and down sinusoidally as it spins.

So Faraday's changing.

Then Faraday's law, Math, Kaling, and Gibe kicks in.

Exactly.

A sinusoidally changing flux induces a sinusoidal EMF.

If Philoduli varies like cosine, the EMF varies like sine, where Montadil is proportional to omega t, where bubble ed is the rotation speed.

That's your AC voltage.

And that drives an alternating current.

And importantly, the electrical power you get out doesn't come from nowhere.

You have to do mechanical work to keep that coil spinning.

The magnetic field exerts forces, torques that resist the rotation, and you have to overcome that resistance.

Energy is conserved.

Okay.

So we've seen induction from moving wires and changing fields.

Now let's shift focus a bit away from the fields in space and more towards the properties of the circuits themselves.

What happens when changing currents interact?

Let's start with two circuits affecting each other, mutual inductance.

Right.

Mutual inductance, usually written as Math Frank, is all about coupling.

If you have, say, two coils of wire near each other, coil one and coil two.

If you change the current flowing through coil one, island one, that changing current creates a changing magnetic field.

And that changing field spreads out.

And some of its flux links through coil two.

So coil two sees a changing magnetic flux, and by Faraday's law, an EMF, Mathio two, gets induced in it.

So the voltage in coil two depends on how fast the current in coil one is changing.

Exactly.

The induced EMF, Math 2 2, is proportional to the rate of change of current.

The proportionality constant is the mutual inductance, Mathrectum.

So Mathrectum di1 dtd.

What determines how big Mathrequin is, the currents?

No, that's the key thing.

Mathrequin depends only on the geometry.

The shapes of the coils, their sizes, how many turns they have, how close they are, their relative orientation, all purely physical arrangement stuff.

It doesn't depend on the currents themselves.

Interesting.

And is the effect symmetric?

Does changing current coil two induce the same kind of EMF back in coil one?

It does.

And remarkably, the coupling coefficient is exactly the same.

Mathrack is always equal to Mathrack data.

That's called the reciprocity theorem.

It's not obvious why it should be true just from looking at coils, but the math works out.

Okay, so that's mutual interaction.

What about a single coil?

Can a changing current in a coil affect itself?

Absolutely.

That's self -inductance, usually denoted by dollar.

How does that work?

Well, any current dollar flowing in a coil creates its own magnetic field, and therefore it's a magnetic flux linking through the coil itself.

Right.

So if you try to change that current dollar, you're changing the coil's own magnetic flux, and by Faraday's law, that changing self -flux must induce an EMF back in the very same coil.

An EMF that opposes the change you're trying to make.

Precisely.

It's called the back EMF.

The formula is mathclo, diddle, s, l, d, w, dine, t.

The dollar is the self -inductance, again, depending only on the coil's geometry, and the minus sign shows it always opposes the change, dado, a, d, t.

That opposition sounds kind of like inertia in mechanics.

Is that a good analogy?

It's a perfect analogy.

Think about mass.

Mass resists changes in velocity, z, y, w, d, t, d, o, dollar, com, acceleration.

It gives things inertia.

Right, five in a dollar.

Well, self -inductance provides electrical inertia.

It resists changes in current.

If you try to increase the current, the back EMF pushes against you.

If you try to decrease the current, the back EMF tries to keep it flowing.

So dollars is like the electrical equivalent of mass.

In many ways, yes.

It measures the circuit's opposition to changes in current flow.

For simple shapes like a long solenoid, you can calculate Lollar directly from its dimensions, number of turns, area, length.

It's a physical property of the device.

Okay.

If Lollar dollar is like electrical inertia, and it resists changes in current, that implies you have to do work against it to get a current flowing, just like you do work to get a mass moving.

Exactly right.

So where does that energy go?

When you've established a current dollars in an inductor, where's the energy stored?

The energy stored in the inductor is given by one dollar, frac one two, L I two two.

Ah, look at that.

It looks just like kinetic energy, one dollar, frac one two, me a D two two.

The analogy holds perfectly.

Dollars like mass, Lollars is like velocity.

The energy you put in to establish the current gets stored, ready to be released if the current changes.

So is the energy stored in the coil, in the wires?

Well, that's where the calculation using dowodor seems to place it.

But Feynman pushes us to think deeper.

Where is the magnetic field?

It's mostly inside the coil for a solenoid, but it also extends outside, filling the space around the wires.

Right.

And the more fundamental view is that the energy isn't really stored in the matter of the wires, but rather it's stored in the magnetic field itself, distributed throughout all the space where the field exists.

How do we connect the L I two two energy, which seems tied to the circuit component, with this idea energy spread out in the field?

There's a formula for the energy density in a magnetic field.

It tells you how much energy is stored per unit volume of space,

and it's proportional to the square of the magnetic field strength B two two.

Specifically, it's frac one B or a frac one C two B two two.

Okay.

If you calculate Bartolors everywhere in space, due to the current dollars in the inductor, and then you integrate that energy density frac B two two over all volume, you get back exactly frac one two two two.

Wow.

So the two views are consistent.

The energy is fundamentally in the field.

That's the deeper physical picture.

Yes.

The energy resides in the configuration of the electromagnetic field in space.

And if we go back to our two couple coils with mutual inductance math frac, how does the energy storage work there?

You just have to account for everything.

The total energy stored is the energy needed to build up the current LR in coil one against its own back EMF, plus the energy for twy dollars in coil two against its back EMF, plus the energy associated with the interaction between them.

So would be.

It comes out to be one dollars frac one two L one I 12 plus frac one two I 22 plus math frac I one I two two.

That last term, the mutual inductance term accounts for the energy stored in the field due to the interaction between the two currents.

Hashtag tag outro.

So wrapping this up, it's been quite a journey through just one chapter, hasn't it?

We've seen that Faraday's law is really two effects in one hat, ultimately unified by the Lorentz force of y dollars equals q plus v times b.

Right.

And we saw how that plays out from the paradoxes like the rotating disk to the incredibly precise engineering of the Betatron.

Yeah, that bad b text orbit a d a condition is just beautiful.

Then we got into inductance LR and math frac as circuit properties measuring electrical inertia and coupling.

And finally, realizing that the energy li two lab two isn't just in the inductor, but is fundamentally stored out there in the magnetic field itself.

It really ties everything together.

Well, thanks for joining us on this deep dive into the fundamentals of induction.

It shows how looking closely at even familiar laws can reveal so much more depth.

Absolutely.

And maybe something for you listeners to ponder.

We saw the energy for two coils is one dollar frac one two l one i 12 plus frac one two plus math frac i one i two two.

Now energy has to be positive, right?

You can't get energy for free.

Right.

It costs energy to build up fields.

Well, the mathematical condition that this energy dollar must always be positive, no matter what currents i dollars and i tidy two you choose forces a purely geometric constraint.

It demands that the mutual inductance math rack can never be larger than the geometric mean of the self

math rack must be less than or equal to score one l and two two.

So the way you can possibly arrange two coils in space is inherently limited by energy conservation.

Seems so.

It's not obvious just by looking at coils, but it's a deep mathematical requirement stemming from physics.

What does that fundamental limit on coupling tell us about the possible geometry of electromagnetic systems?

Something to think about.

Indeed.

Food for thought.

We'll