Chapter 21: Solutions with Currents & Charges

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Welcome back to the Deep Dive.

Today we are taking, well, a pretty massive leap in fundamental jumping straight into Chapter 21 of the Feynman Lectures on Physics Volume 2.

If you've been following classical physics, you know we spent quite a bit of time in the relatively cozy world of electrostatics where charges just sit still.

Yeah, not anymore.

Our mission today is definitely moving into dynamic electromagnetism.

We're tackling the core mechanism really that connects moving charges, accelerating currents, and ultimately electromagnetic waves.

This chapter gives us the complete general solutions to Maxwell's equations when charges and currents are actually doing something moving around.

This is where electromagnetism ties directly into light, radio waves, I mean every form of radiation you can think of.

Okay, that sounds like a bit of a heavy lift.

Maxwell's equations themselves are, let's face it, complex enough.

Solving them in motion seems like it could be a nightmare.

What's the fundamental shift we really need to get our heads around right at the start?

It's all about time.

Or, maybe more accurately, the finite speed of light.

See, in statics, we basically assume the field just is there, instantaneously.

But if we wiggle an electron right here, the change in the field it generates doesn't magically appear way over there at the same moment.

It has to travel, and it travels at speed c.

This forces us to use the concept of retardation.

Retardation meaning like a delay.

Exactly, a delay.

The field you observe right now, at this time t, isn't caused by what the source charge is doing right now at time t.

No, it's caused by the charge's position, its velocity, its acceleration, and an earlier time.

We call this the retarded time, and you calculate it pretty simply.

It's just t minus r over c, where r is the distance and c is the speed of light.

That rc is just the travel time.

Okay, right.

Like if we were looking at a star 100 light years away, we're seeing the light that left 100 years ago.

So we're seeing the star as it was.

Same idea here.

Precisely the same idea.

This little adjustment, thinking in terms of t minus r over c, it's absolutely central.

You know, the fundamental guarantee of causality and classical electrodynamics.

Everything depends on what the source was doing back then at the retarded time.

Okay, got it.

So let's dive into the first big solution Feynman gives us.

It's for the electric field, the E field, produced by just a single charge moving in any which way.

It looks incredibly complicated on the page, but he breaks it down into three distinct parts, right?

Three physically meaningful components.

Oh yeah, that equation for E is notoriously long, but it's brilliant because those three terms really capture the physics, the difference between what's happening close to the charge versus far away, the radiation field.

We're calculating the electric field at some observation point based on the charge's position in motion, but crucially all evaluated at that earlier retarded time.

Right.

Term one seems the most familiar.

He calls it the retarded Coulomb field.

Exactly.

That's sort of the static anchor.

It looks just like the good old one over r squared dependence.

We know from Coulomb's law,

but the key difference is we use the charges position at the retarded time, not its current position.

This term is what really dominates the field when you're very close to the charge.

Then there's term two.

It looks like a, a pretty complicated correction factor.

It involves the rate of change of that first Coulomb like term.

That's right.

It's necessary for mathematical completeness, but it doesn't fundamentally change the long distance story because it also falls off like one over r squared.

Okay.

But then then we get to term three, the acceleration term, and this is where things get really interesting, isn't it?

This is absolutely the star of the show.

This term depends entirely on the acceleration of the charge and unlike the first two, it falls off much more slowly only is one over r just one over r one over r versus one over r squared.

That difference sounds small, but physically it's huge, right?

Oh, it's monumental.

Think about how fields decay with distance.

One over r always beats one over r squared.

Eventually.

Imagine you double the distance are one, the one over r squared field strength drops by a factor of four, but the one over our field strength, it only drops by a factor of two.

So if we just step far enough away from this moving charge, those first two terms, the sort of near field stuff, they basically vanish compared to the third term.

And the only field left, the only thing carrying energy outwards indefinitely is that third term, the acceleration term.

Correct.

That one over our term is the electromagnetic radiation.

It doesn't matter if we're talking about a tiny flicker of light or a huge radio wave.

If a charge accelerates, and remember that means changing speed or direction it has to shed energy that propagates outward potentially forever.

The static like fields dominate up close, but it's the acceleration fields that carry energy far away.

Okay.

I think I'm following, but let me ask this.

You said the one over r squared term is much larger close up.

So how can we be so sure that only the one over our term actually carries energy away in the long run?

And what's the physical distinction there?

That's a really good question.

Bulls down to energy flow.

Those one over r squared terms, they represent fields that are fundamentally tied to the source.

They might fluctuate.

Sure.

But they don't represent a net permanent outflow of energy is more like energy sloshing back and forth near the charge.

But the one over our fields, well, when you actually calculate the flux using something called the pointing vector, those fields show a clear net outward flow that keeps going even at enormous distances.

That decay rate one over r is the signature of energy truly escaping.

That makes sense.

The energy flow picture clarifies it.

Okay.

So that complex three -part equation was for just one single point charge moving arbitrarily, but to really tackle dynamic electromagnetism generally, you know, like current in a wire or a whole blob of charge moving around, we can't just sum up that super messy equation for every single point.

We have to switch gears, right?

Use potentials.

Exactly.

When you have charge density, row and current density, J spread out over some volume, trying to solve directly for E and B using that point charge formula becomes, well, geometrically impossible, practically speaking.

It's far, far simpler to first solve Maxwell's equations for the helper quantities, the scalar potential phi and the vector potential A.

Right.

And the beauty here, as Feynman points out, is that Maxwell's equations can actually be rewritten as pretty straightforward wave equations for these potentials phi and A.

They absolutely can.

And the form of the solution immediately tells you about waves.

For, say, a point source generating some potential Ci,

the solution has to be proportional to some function F evaluated at t minus r over c, all divided by r.

That t minus r over c part explicitly confirms disturbance spreads out like a wave at speed c, and the 1 over r factor concerns it's a spherical wave, weakening as it expands.

So stepping back just a bit, how do we combine this idea of retardation, the t minus r over c and this wave spreading into a general solution for, you know, continuous distributions of charge and current, not just a point source?

This is where one of the most powerful principles in physics comes in handy, superposition.

We just treat the entire volume of charge and current as if it's made up of an infinite number of tiny, point sources, each in a little volume element dV.

The total potential we measure at our observation point is simply the sum, or rather the integral, of all the individual contributions from every single one of those tiny volume elements, each calculated using its own retarded time.

And this approach leads directly to the big general solutions presented in the chapter, right?

The scalar potential phi is basically the integral over the whole volume of the charged density rho, but evaluated at the retarded time for each bit of charge.

And similarly, the vector potential a is the integral over the retarded current density.

That's precisely it.

These integrals built entirely on the retarded sources are the complete fundamental description of how any distribution of moving charges and currents generates electromagnetic fields.

Once you manage to solve these integrals to find phi and a, which can still be tricky, mind you then just take their derivatives, their spatial gradients and time derivatives, and out pop the ENB fields we actually observe.

It's a complete dynamical picture.

Okay, let's try to apply this powerful machinery.

Let's make it concrete with a classic example Feynman uses, the oscillating dipole.

This is like the simplest possible model for an antenna, or maybe even how an atom might radiate light.

So we imagine two equal and opposite charges really close together, separated by a tiny distance d, and they're just wiggling up and down sinusoidally.

Exactly.

These oscillate charges create an electric dipole moment key that oscillates like a sine wave.

That oscillating p is our source.

So we take this oscillating source term and we plug it into those general retarded potential integrals we just talked about.

And what's really neat is how the resulting fields behave depending on how far away you are.

Feynman shows that near the dipole the fields are kind of a complex mess, a mixture of those near -field one over r squared terms and the radiation one over r terms.

But what happens when you get far away out in the radiation zone?

Right, at large distances the messy near -field terms just die away relatively quickly.

The fields simplify tremendously.

What you find is that the dominant magnetic field, b, is proportional to one over r, and crucially the dominant electric field, e, is also proportional to one over r.

There it is again, the one over r dependence.

That's the proof in the pudding, isn't it?

It confirms that insight we got from the single accelerating charge.

The oscillation, which is just continuous acceleration and deceleration, means the dipole must radiate energy, energy that travels far away, falling off only as one over r.

Absolutely.

And not only that, the calculation actually shows you how it radiates.

The fields end up being strongest perpendicular to the direction the charges are wiggling.

And the e and b fields are perfectly perpendicular to each other and also perpendicular to the direction the wave is traveling.

They're exactly like transverse light waves.

You can almost see the radio wave being born right there in the math.

Okay, amazing.

Now, there's one final crucial piece to this puzzle, focusing on getting things extremely accurate, especially for fast -moving charges.

The general solution used those smeared -out densities, rho and j.

But what if we want the exact precise solution for a single point charge, q, especially if it's moving really fast, maybe even close to the speed of light?

Yes, that requires digging a bit deeper into the math, leading us Leonard Weicher potentials.

These are essentially the specialized exact solutions you get when you carefully evaluate those general integrals, specifically for a point charge q.

And what's fascinating is that when a charge is moving very fast, just using the retarded distance, r retarded, in the simple Coulomb formula isn't quite enough.

There's a subtle correction needed.

Wait, why isn't it enough?

We already accounted for the signal delay using the retarded time, t minus r over c.

What else needs correcting?

It turns out we need to account for how the charge's own motion during the light travel time kind of bunches up or spreads out the potential it seems to be emitting from the observer's point of view.

So the scalar potential phi isn't just q divided by 4 pi epsilon not times the retarded distance anymore.

There's a crucial correction factor that appears in the denominator.

It's 1 divided by 1 minus the dot product of r and v over c.

Okay, whoa, that velocity term v in the denominator,

that looks complicated.

Can you give us a feel for what that denominator factor is actually doing physically?

What's the intuition?

Well, think about maybe the Doppler shift from an ambulance siren.

If the ambulance is racing towards you, the sound waves get compressed.

You hear a higher pitch.

It's a similar idea here, but for the potential field.

If the source charge is moving rapidly towards your observation point, the effective time interval between successive packets of field reaching you gets squished, mathematically speaking.

It looks denser.

If it's moving away, it looks stretched out.

That denominator factor, the 1 r v c term,

precisely accounts for this geometric and temporal distortion caused by the source's velocity relative to you, the observer.

Ah, okay.

So it's correcting for how the moving source effectively sweeps over the observation point, making the charge seem denser or sparser depending on its velocity towards or away from you.

Exactly.

It's a subtle but critical correction for accuracy, especially when the velocity v gets close to c.

Right.

And this leads us directly to one of the most structurally profound moments in the entire chapter, maybe in all of classical E &M.

When Feynman looks at the special case of a charge moving at constant velocity v, so no acceleration, just steady motion, and solves the Leonhard Weitcher potentials for that specific case,

what falls out?

What falls out is amazing.

The resulting scalar potential 5L explicitly contains a factor that should make any physicist's jaw drop 1 over the square root of 1 minus v squared over c squared.

That's the Lorentz factor, gamma, the exact factor that defines time dilation and length contraction in Einstein's special relativity.

It just appears.

It just appears, naturally.

This isn't some coincidence or approximation.

It's rigorous mathematical proof that Maxwell's equations formulated way back in the 19th century, long before Einstein, were already perfectly consistent with the weird rules of special relativity.

The dynamics of electricity and magnetism didn't need fixing for relativity.

They already had it built in.

It was Newtonian mechanics that needed the overhaul.

Incredible.

So looking back at this whole chapter, this whole deep dive, what really stands out to you?

We've gone from the field of a single accelerating charge through the general potential integrals, saw the oscillating dipole radiate, and ended up with this relativistic correction from Lienard and Weitcher.

For me, it's the completeness and the internal consistency of the structure Maxwell built.

It's just beautiful.

The key conceptual takeaway has to be retardation, that at minus rho over creo, which ensures causality, cause always preceding effect.

And tied to that is the inescapable conclusion that the acceleration of charge represented by that 1 over return in the fields is the fundamental engine of all electromagnetic radiation.

Every antenna, every light bulb filament, every photon emitted by an atom, fundamentally relies on that principle,

accelerating charge radiates.

It really ties everything together.

It's astonishing that this whole elegant framework was basically sitting there fully formed before Einstein even started thinking about relativity.

So here's a final provocative thought for all of you to maybe chew on.

The fact that the potential for a simple charge moving at constant velocity already contained that iconic relativistic factor, gamma, suggests something profound.

It means the laws of electricity and magnetism were speaking the language of relativity long before we explicitly formulated it.

So the question is, was the universe trying to tell us that space and time were relative through the behavior of electric and magnetic fields all along?

We'll leave you to ponder that one.