Chapter 27: Field Energy & Momentum in Electromagnetism

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Welcome to the Deep Dive.

Today our mission is, well, it's pretty fundamental.

We're looking into how energy and momentum work, specifically where they actually are and how they move in the electromagnetic field.

That's right.

It's a concept that really challenges the sort of classical way of thinking.

We're moving beyond seeing energy and momentum as just properties of particles.

And we're using a classic source for this, right?

Feynman's lectures on physics.

Exactly.

Volume 2, chapter 27.

Feynman forces us really to see the electromagnetic fields, the E and B fields,

as physical things, things that hold energy, hold momentum.

Okay.

So we're going step by step through his derivation of these laws, focusing on this idea of local conservation.

Local conservation, yes.

That's the key.

So our goal, as always, is to make the math, which can be pretty dense, feel intuitive, get that Feynman -esque feel.

So let's start with conservation itself.

What's the big deal there?

Well, most people, when they think about conservation, like conservation of charge, they're thinking about what Feynman calls worldwide conservation.

Meaning the total amount of charge in the whole universe stays the same.

Precisely.

If some charge vanishes over here, an equal amount must instantly appear somewhere else, even millions of miles away.

The total count is always right.

But that instant action, that's a problem for physics, isn't it?

Especially with relativity.

Exactly.

Relativity tells us nothing can happen instantaneously across a distance.

If charge leaves point A and appears at point B, it has to travel through the space between A and B.

It takes time.

So that worldwide view just isn't enough.

Right.

We need local conservation.

This is much stricter.

It says if the amount of something, charge, energy, whatever, decreases in some small volume of space,

then it must be because it flowed out across the boundary, the surface of that volume.

Like water leaving a leaky bucket.

You see the level drop because water is flowing out through the hole.

That's a great analogy.

The change in the density inside is directly linked to the flow across the surface.

Mathematically for charge, the rate of decrease of charge density, down O, has to equal the divergence of the current density A in dollars.

Okay.

So density goes down, current flows out.

That makes sense.

And this local idea is what we need for energy in the electromagnetic field.

Yes.

If energy is conserved locally, we need to define two new things for the field itself.

First, an energy density.

Let's call it meter.

That's how much energy is stored per unit volume right there in the field.

Like how much energy is packed into a cubic meter of space just because there's an E or B field present?

Exactly.

And second, we need an energy flow vector, usually called zeal dollars.

This tells us how fast energy is flowing across a unit area and in what direction.

It's the rate of energy transport.

So lies the storage, side is the flow.

And the goal is to find a conservation law that links them.

Right.

Pointing's theorem is basically that law.

It says that the rate at which the total field energy inside a volume decreases, plus the rate at which energy flows out through the surface, that's related to zeal dollars, must equal the rate at which work is done on the matter inside that volume.

Work done on the matter, like heating up a resistor.

Precisely.

Or accelerating charges.

It's an energy accounting equation.

Energy lost by the field either flows away or gets transferred to matter locally.

And the amazing part you mentioned is that the math, starting from Maxwell's equations, it gives us formulas for $2 dollars dollars that only depend on the E and B fields themselves.

That's the beautiful result.

It proves the energy isn't just associated with the charges, it's in the field.

The field itself is the carrier and store of energy.

Okay.

So what are these formulas?

Conceptually, what do they tell us?

Let's start with the energy density.

Okay.

$2.

It turns out to be the sum of two terms.

One part depends on the electric field squared, E22, and the other part depends on the magnetic field squared, T2.

Specifically, it's epsilon dollars E22 plus B2 dollars.

So energy density goes up with the square of the field strength.

Stronger fields pack way more energy into the same space.

That's right.

It's stored in both the electric and magnetic components of the field.

And now the big one, Dillers.

The energy flow vector.

The pointing vector.

Yes, the pointing vector.

$7.

This one tells you where the energy is going and how fast.

The derivation shows, save in a dollars is equal to $1 times the cross product of V in a dollars.

Yeah.

So, seven al -Qaeda.

T times B.

The cross product.

E times B.

That means the energy flow direction is perpendicular to both the electric field and the magnetic field.

Exactly.

That perpendicular relationship is key, and it leads to some really non -intuitive results when you look at specific examples.

Before we get to those, you mentioned something about ambiguity, that these formulas aren't the only possibility mathematically.

Ah, yes.

That's an important subtlety, Feynman points out, because the conservation law involves derivatives.

Mathematically, you could add certain other terms to TWAN and Zivildal specifically.

Terms related to the divergence of some arbitrary vector field, and the overall conservation equation would still hold.

So, how do we know these specific formulas, like one -all times BR are the right ones?

Good question.

We stick with these forms.

One -all -all dollars, epsilon E22 plus B2 dollars, E times B million for a few reasons.

One, they're the simplest forms that pop out directly from Maxwell's equations.

Two, they make physical sense in terms of energy localization.

And three, critically, they work.

Every experiment involving radiation pressure, energy transport by light, antenna radiation, it all confirms these expressions for two and a dollar dollars.

Okay.

So, physics chooses the simplest experimentally verified solution.

Makes sense.

Now, let's use this pointing vector, I or E, times B to visualize energy flow.

This is where it gets weird, you said.

It can, yeah.

Let's start easy.

A light wave traveling in empty space.

You have an electric field, E oscillating, say, vertically.

You have a magnetic field, B oscillating horizontally.

They're perpendicular.

Right.

And the wave travels forward, perpendicular to both E and B.

Okay.

Now, take the cross product, we comes B and U.

If E is up and D is right, mu di O times B di O points forward.

Yeah.

Exactly in the direction the wave is moving.

So, for a light wave, points in the direction of propagation.

The energy flows along with the wave, that feels right.

It does.

The energy density dollar travels along with the wave carried by the flow, carried by the flow.

The average energy flow is related to the average E to our time.

No surprises there.

All right.

Now, for the counterintuitive stuff.

Charging capacitor.

A simple parallel plate capacitor.

Okay.

You connect it to a battery.

Current flows, charge builds up on the plates.

An electric field, E, builds up between the plates, pointing from positive to negative.

Let's say E points downwards.

And energy is stored in that E field.

Right.

That's the other part.

But how does it get there?

The solar part.

Think about the B field.

While it's charging, there's a current flowing in the wires connected to the plates.

A current creates a magnetic field that circles around the wire.

Okay.

Now, consider the space just outside the gap between the plates.

You have the E field pointing, say, downwards, or fringing outwards slightly.

And you have the B field circling around that gap region caused by the effective displacement current as the E field changes.

Or just thinking about the wires feeding it.

Right.

B is circling horizontally around the vertical E field.

Now, do the cross product.

E times B1.

If E is down and B is circling, say, counterclockwise when viewed from above, 1 times BD points.

Radially, inward, towards the center of the capacitor gap.

Exactly.

The pointing vector zero points inward from the sides, from the space around the plates,

towards the region where the E field is building up.

Whoa.

So the energy doesn't flow along the wires and onto the plates?

Nope.

The wires guide the charges, setting up the E field.

But the energy itself flows into the gap from the surrounding space, carried by the interacting E and B fields in that space.

That completely changes how I picture charging a capacitor.

The energy comes in sideways.

It does.

It's a beautiful illustration of the field carrying the energy.

Let's take another one.

A simple wire carrying a current, getting hot, jewel -eating.

Okay.

A resistor.

Energy is being dissipated as heat.

Power is lost.

Passer PLEs pee off 8 to our dollars.

Where does that energy come from?

I always assumed it flowed down the wire like water in a pipe.

That's the intuitive but incorrect picture, according to field theory.

Think about the fields.

There's a current dollar slowing, so there's a magnetic field dollar circling the wire.

Right.

Ampere law.

And because the wire has resistance to push the current through it, there must be a small electric field in Euler pointing along the wire, in the direction of the current flow.

Okay.

E is along the wire.

B circles around it.

Now, one Euler times B.

Okay.

Use the right -hand rule.

If E is pointing along the wire and B is circling it, the cross product, z dollar, secues E times B meter one points.

Perpendicular to the wire, pointing inward, towards the center of the wire.

Precisely.

The energy that's being dissipated as heat inside the wire is flowing into the wire from the electromagnetic field in the space surrounding the wire.

So the power line's outside my house.

The energy isn't really inside the copper.

It's flowing in the space between the wires, guided by them, and then flows into my appliances from the surrounding space.

That's what the pointing vector picture tells us, yes.

Yeah.

The wires act as guides for the field, and the field carries the energy, delivering it perpendicularly into the resistive loads.

It's quite a paradigm shift.

It really is.

And there was one more example about static fields.

A charge near a magnet.

Ah, yes.

This one's subtle.

Imagine a stationary point charge and a stationary bar magnet nearby.

The E field from the charge spreads out radially.

The B field from the magnet loops around.

Both are static.

Nothing's changing.

Okay, so no energy transfer, presumably.

No net energy transfer, no work being done.

But both E and B exist in the space around them.

So you can calculate z dollar times B me motoro.

And it's not zero.

It's generally not zero.

If you map it out, you find the pointing vector circulates and circulates.

It flows in closed loops in the space around the charge of the magnet.

Energy is flowing, but in circles.

What does that even mean?

It means there's a constant circulation of energy in the static field.

No energy is being gained or lost overall, but it's constantly flowing around.

It reinforces the idea that the field is a dynamic entity, constantly active, even when things look static macroscopically.

Wow.

Okay, so fields store energy and transport energy.

Right.

That's established.

But if they carry energy, and especially if that energy moves at the speed of light, relativity must connect this to momentum, right?

Absolutely.

That's the next logical step Feynman takes.

If the field has energy density dollars and energy flow dollars, it must also have momentum.

We define a field momentum density.

Let's call it do dollars.

This is momentum per unit volume stored in the field.

And how does it relate to energy?

Relativity gives us the link.

We know that energy dollars has an equivalent mass, maybe of two to two, and moving mass has momentum.

For light, the momentum two dollars is related to energy one dollar by two cedar.

More generally, Einstein showed that energy flow itself implies momentum.

Okay.

The relationship turns out to be beautifully simple.

The momentum density dollars is directly proportional to the energy flow vector, seven dollars plus, specifically delphi one two.

So dollars equals divided by the speed of light squared.

Wherever energy is flowing, there's momentum associated with that flow, pointing in the same direction.

Exactly.

If dollars is the energy crossing a unit area per unit time, then Zefi 22 is the momentum crossing that same unit area per unit time.

And we see this, like light pushing things.

We do.

Radiation pressure is the direct experimental proof.

When light hits an object and gets absorbed or reflected, it transfers momentum to that object.

Think of solar sails.

The tiny push from sunlight is due to the momentum carried by the electromagnetic field of the light wave.

So the field is carrying real mechanical momentum.

It's not just an abstract idea.

It's entirely real.

And this means we have to update our conservation of momentum law.

The total momentum of a system isn't just the sum of the momentum, all the particles, some mbf.

Ah, you have to add the momentum stored in the field too.

You have to add the integral of the momentum density dollars over all space.

So total momentum, sum of particle momentum plus total field momentum, and that total quantity is what's conserved locally.

So if two charges repel each other and fly apart, gaining mechanical momentum, the field between them must lose an equal amount of momentum to keep the total constant.

Precisely.

The field acts as a reservoir or a source sink for momentum, just like it does for energy, ensuring local conservation holds for the complete system particles plus fields.

And this extends even to angular momentum, doesn't it?

Feynman had that final kind of puzzling example.

He did.

It's a bit more complex, involving a spinning charged object near a magnetic field source, like a solenoid.

If you turn off the current in the solenoid, the B field disappears.

The calculation shows that as the B field collapses, it induces an E field, which puts a torque on the charged object, causing it to start spinning.

It gains angular momentum.

Where did that angular momentum come from?

Linear momentum seems conserved, but suddenly there's rotation.

The only explanation that works is that the initial static electromagnetic field, the combination of the E field from the object and the B field from the solenoid, must have been storing angular momentum.

When the B field collapsed, that stored field angular momentum was transferred to the object.

So even a static field configuration can hold angular momentum.

Yes.

It's mind -bending, but necessary to make conservation of angular momentum work locally.

It really cements the idea that the field isn't just some population tool.

It's a physical entity with energy, linear momentum, and angular momentum.

Which means worldwide conservation only truly makes sense if you include the field as part of your world.

That's the essential conclusion.

The fields are part of the machinery of the universe, participating fully in the conservation laws.

Okay, let's try to wrap this up.

A quick recap of the big ideas from this deep dive.

We went from simple worldwide conservation to the much stricter requirement of local conservation.

And that forced us to treat the electromagnetic field itself as a physical thing.

We found it stores energy with a density dollars, depending on U2 tool and or B tool it loves.

And it transports energy governed by the pointing vector zeodollary equals E times B mugere, which shows energy flows in surprising ways sideways into capacitors, inwards into resistors.

Right.

Then, connecting energy flow to momentum via relativity, we found the field also carries momentum density, d dollars Se2, which explains radiation pressure.

And finally, even angular momentum can be stored in static fields, highlighting just how physical and dynamic these fields are.

The key takeaway is that conservation laws for energy momentum and angular momentum are only complete when you account for the contributions stored and transported by the fields themselves.

It really changes everything.

Thinking about this, if the field carries energy momentum, maybe even angular momentum, and it mediates all these interactions,

what does that really say about empty space?

It's certainly not empty, is it?

Not at all.

It's teeming with potential, capable of storing and transmitting the fundamental quantities of physics.

It's the stage and part of the action.

Definitely something profound to think about as you continue exploring electromagnetism.

A perfect thought to leave our listeners with.

Indeed.

Well, thank you for joining us on this exploration of field energy and momentum based on Feynman's insights.

We hope it sheds some light on these crucial concepts.