Chapter 14: The Magnetic Field in Various Situations

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Welcome back to the Deep Dive.

Today we're really getting into the core machinery of electromagnetism.

We are indeed.

We're tackling a pivotal chapter diving deep into how we handle magnetic fields, specifically with a powerful new tool,

the vector potential,

usually called a baller.

That's the one.

Think back to electrostatics.

Calculating the electric field, that's often way easier if you find the scalar potential to feel first, right?

Right.

Find a fail, take the gradient, boom, you have a baller.

Much simpler sometimes.

Exactly.

Now, for steady currents and the magnetic fields they create, the dollar field,

well, it's more complex.

Currents are vectors, the fields loop around.

You need a vector tool to simplify things.

And that tool is this vector potential.

Precisely.

It brings back that kind of mathematical elegance we found with LRI.

Okay, but why do we need it?

I mean, can't we just use Ampere's law or Biot -Savart?

Why invent something new?

Is it just for convenience or is there a deeper reason?

Ah, well, it's both, but the deeper reason comes first.

It's tied to a fundamental property of magnetic fields, at least in magnetostatics for currents or study.

Okay, the key physical fact is,

there are no magnetic monopoles, no isolated north or south poles analogous to positive and negative charges.

Right, you always get dipoles, loops, you cut a magnet in half, you get two smaller magnets.

Exactly.

And mathematically, this translates directly to saying the divergence of the magnetic field is always zero.

NABLA DOB equals zero deal.

Everywhere.

Which is the math way of saying the field lines always close on themselves, no beginnings or ends.

Perfect.

That's exactly what NABLA BODD means physically.

And here's the mathematical magic.

Any vector field whose divergence is zero can always be expressed as the curl of some other vector field.

Always.

That's a strong statement.

As a fundamental theorem of vector calculus, if NABLA AD equals B, then there must exist a vector field 8 such that BALA E is equal to the curl of A.

So BALA A times H A A TASME 2.

That's basically the definition of a dollar.

Okay, so A dollar comes from the gradient of the scalar R and a dollar comes from the curl of the vector.

There's a nice parallel there.

There really is.

The curl operation, it measures the rotation or circulation in a field.

Since battlefield lines circulate around currents, it kind of makes intuitive sense that battle dollar is related to the circulation of dollar.

So finding out a dollar first turns the problem of finding the dollars into something potentially easier.

That's the hope.

Find out a dollar, then compute its curl.

Now you called a mathematical tool.

Is it unique?

If I know dollars, is there only one possible dollar that could have created it?

Great question.

And the answer is no, it's not unique.

This is a really important point.

Remember how adding any constant to the scalar potential E field dollar didn't change the electric field because the gradient of a constant is zero?

Yeah, the potential difference is what matters, not the absolute value.

Right.

Well, for all of there's something similar, but a bit more complex.

You can add the gradient of any arbitrary scalar field, let's call it DC to DR.

So you get dollars, a dollar plus nobly.

If you then take the curl of this new dollar, you get nabla times a plus nabla times a plus nabla times, but the curl of a gradient is always zero.

That's another vector identity.

Ah, so the extra term vanishes.

Exactly.

So by dollars, nabla times a equals nabla times a.

You can change dollars in this specific way by adding a gradient without changing dollars at all.

This freedom to choose is called gauge freedom.

Gauge freedom.

It sounds like physicists just gave themselves a way to make the math easier by picking the right gauge.

Well, yes, that's exactly what we do.

It's like choosing where zero elevation is.

You can set it at sea level or the ground floor, whatever is convenient.

The height difference between two points stays the same.

That's like by a dollars.

The absolute potential dollar depends on your arbitrary zero point, your gauge choice.

So what's the convenient choice here?

In Megidda statics, we often choose the coulomb gauge.

It's defined by setting the divergence of 80 zero nabla dot a equals elite knot.

This choice simplifies the equations we need to solve later on.

Okay, that feels a bit abstract.

Let's try to visualize.

Soller, if we have a simple uniform magnetic field, say it's a dollar pointing straight up along the z -axis,

what does the other field that produces that dollar look like?

Is it just pointing up to?

Surprisingly, no.

And this is key to understanding that dollars isn't just dollar in disguise.

For a uniform baller field pointing along z, the vector potential dollar actually circulates around the z -axis.

It circulates like in circles.

Yes, in the side plane, the magnitude of dollar increases linearly with a distance from the z -axis and its direction is tangential to circles centered on the z -axis.

Think of stirring water in a bucket.

The water circulates.

So baller points up, but dollar goes around.

That seems weird.

Why?

Because baller isn't dollar.

It's the curl of a dollar.

The curl maybe is how much a field is circulating or rotating locally.

To get a constant upward dollar field, you need a dollar field that's constantly circulating around that axis with its circulation strength increasing as you move outwards.

Okay, I think I'm starting to see.

A dollar reflects the potential for circulation that results in a dollars.

It's definitely a distinct field.

It is.

It carries information about the sources in a different way than baller does.

All right, so we accept all or exists.

It's related to dollar by curl.

It's not unique, but we can choose a gauge like novel RDA equals dot.

Yeah, the big question.

How do we actually find all or if we know the currents dollar that are creating the magnetic field?

Right, the practical part.

We start with Ampere's law for magnetostatics.

A novel times B equals the wood.

Oh, it finally uses Gaussian units.

So it's novel times B, fornthic J, and we substitute blood at times all at times A.

So you get novel times novel times A, that looks complicated.

Curl over curl.

It does, but there's another vector identity, novel times novel times a nabalade.

That first term, nabalata A, just becomes zero because we chose the coulomb gauge.

Brilliant simplification.

Okay, so that leaves nabalatelda A, but just becomes zero because we chose the coulomb gauge.

Brilliant simplification.

Okay, so that leaves nabalatelda A for plex J, or novel 2A equals four plex J.

Almost.

In Feynman's units, most one is absorbed differently, and the equation becomes nabalatelda A equals J epsilon C2.

That's the form analogous to electrostatics.

Let me see.

The equation for the scalar potential file from charge density order was Poisson's equation.

Nabalatelda A equals two A.

Excellent.

Look at the structure.

Wow, they are mathematically identical.

Just replace the scalar phi with the vector A equals the scalar charge density, or is with the vector current density, and that two to two factor.

Exactly.

The Laplacian operator, nabalatelda A, acting on the potential, equals the source density times constants.

This is huge.

Why huge?

Because we already know how to solve Poisson's equation.

Precisely.

We spent ages solving nabalatelda phi as well as epsilon dollars in electrostatics.

The general solution is an integral.

A phi at some point is the integral of devour over all space divided by the distance.

So we can just write down the solution for to buy analogy?

Yes.

Since nabalatelda A nashay, epsilon C2 is just three scalar Poisson equations, one for x with source Jx, one for A with J, or from errors with JsA, the solution for each component of Aad is the same kind of integral.

You mean x is found by integrating Js over volume divided by distance, A is by integrating J, and A is by integrating JDs.

That's exactly it.

One mathelon CAADDBR.

It replaces trying to solve the Biot -Savart law, which involves a tricky vector cross product inside the integral, with three potentially much simpler scalar integrals.

Okay, that is a significant computational advantage.

Let's see it in action.

First classic example.

The long straight wire carrying current A to J.

Okay.

Assume the wire is along the z -axis.

The current density J only has a z -component, JzD.

Should we only need to calculate A's right?

epsilon A will be zero because J was in J or zero.

Correct.

We solve the nabalatelda epsilon CD.

This is mathematically identical to finding the electric potential DDA from a long line of charge.

And the solution for that was proportional to the natural logarithm of the distance from the line.

Exactly.

So Azor is proportional to AR.

Then to get A $, we compute Adalible times AR.

Since Adal only has an Azor component that depends on tier R, the Krull calculation gives A $ field that points purely tangentially, circling the wire, and its magnitude falls off as O $.

Which is the known result for a long wire.

Okay, the method works, checks out.

It confirms the machinery.

But as you said, not earth -shattering news there.

So does this Adal field ever give us something genuinely surprising, something that Badal alone doesn't show us?

Yes, absolutely yes.

The long solenoid, this is where dollars starts to feel,

well, more real perhaps.

The solenoid.

Okay, that's the coil like a spring carrying a surface current.

What's the standard result for Badal dollars?

They're an ideal infinitely long solenoid.

The magnetic field dollars is perfectly uniform inside the coil, pointing along the axis.

And it's exactly zero everywhere outside the coil.

Right.

B $ is outside.

Put a compass out there, it doesn't move.

Now, let's calculate the vector potential dollars for this solenoid, using our integral formula, or by solving the differential equation.

Okay, what do we find?

Inside, we find an Olery field that circulates around the axis, similar to the uniform field case we discussed earlier.

Which makes sense, because Badal dollar is uniform inside.

But outside?

Outside, a Baller is zero.

So Baller must be zero too, right?

Or at least constant.

Because its curl has to be zero.

Ah, but remember, a dollar doesn't have to be zero for its curl to be zero.

It could be the gradient of some scalar function.

And in this case, outside the solenoid is not zero.

Wait, seriously, V $ is outside, but 1 $ outside, but 1 $ outside.

Correct.

Outside the solenoid, still circulates around the axis, but its magnitude falls off as non -zero.

Even though the local magnetic source field, the dollar, is identically zero.

Hold on, how can that be?

If a dollar is just a mathematical tool to get Badal dollars, how can it exist where Baller doesn't?

What does it mean for dollar to be non -zero there?

That's the profound point.

It means odd -odders contains information that Baller doesn't.

At least not locally.

The non -zero dollar outside is related to the total magnetic flux trapped inside the solenoid.

Even though you can't measure a dollar field locally outside, the system knows about the trapped flux via the dollar field.

So dollar has a kind of non -local property.

It feels the effect of the dollar field somewhere else.

In a sense, yes.

And this has real physical consequences, most famously demonstrated in the Aharonov -Bohm effect in quantum mechanics, where charged particles are affected by odd -odder, even when moving through regions where Badal dollars dollars.

It strongly suggests A $ loss is perhaps more fundamental than dollar.

Wow.

Okay, that is surprising.

Let's shift gears slightly.

What about localized sources, like a tiny loop of current?

Yes, this leads us to the magnetic typo moment.

If we take a small rectangular loop, say side lengths and current dollars, and we calculate the vector potential dollar far away from this loop.

Using the integral Juncture DVV again?

Yes.

Approximating for large distances.

We find that the Oller field looks like the potential from a magnetic dipole.

We define the magnetic dipole moment, Myrto.

Its magnitude is the current times the area of the loop.

Okay, Myrto i times area.

And it's a vector.

Which way does it point?

Perpendicular to the plane of the loop, usually given by a right -hand rule.

Curl fingers in direction of current, thumb points in direction of view.

Got it.

So we find, far from the loop, what about the Boller field that comes from this S?

When you calculate Boller times,

using the expression for dollar from the magnetic dipole, you find something remarkable.

The resulting Boller field components have exactly the same mathematical form as the electric field LR components far from an electric dipole.

An electric dipole.

You mean like a small separation between a positive and a negative charge.

Exactly.

The field patterns look identical at large distances.

One generated by a tiny current loop, the other by separated charges.

That's a striking similarity.

But you emphasized earlier, no magnetic charges.

Absolutely crucial distinction.

The source is fundamentally different.

The magnetic field comes from circulating current, quarter times area, while the electric field comes from separated static charges, quarter times distances.

Even if our fields look the same mathematically, we must remember the magnetic field originates from moving charge, not static magnetic poles.

Okay, important caveat.

Now, one more connection.

Most of us probably learned the Biot -Savart law first for calculating Biot -Savart law first, for calculating biot dollars from currents.

How does that fit into this whole vector potential picture?

Is it related or separate?

Oh, it's directly related.

The vector potential method is arguably more fundamental.

The Biot -Savart law can be derived from the vector potential.

Really?

Remember our integral solution for epsilon?

For currents flowing in thin wires, the current density data times the volume element, 20 phi, can be replaced by the current dollar times the path element vector, math bfs.

Okay, so the volume integral becomes a line integral along the wire.

Propto infracto inch.

Precisely.

Now, if you take this expression for all dollar, which is a line integral, and you calculate ball dollars per meibola times a, well, the vector calculus gets a bit involved.

You have to be careful with the curl operating on the integral.

But when the dust settles?

When the dust settles, you get exactly the Biot -Savart law for the magnetic field ball dollar.

Wow.

So Biot -Savart is just what happens when you find a dollar for a wire first and then take its pearl.

Essentially, yes.

It shows the vector potential framework is the deeper one.

So this brings us back to the practical side.

If Biot -Savart gives ball a dollar directly,

why did we go through all this trouble with auto rolls?

You said it's often easier.

It often is, yes.

Let's recap why.

Finding ball a dollar with Biot -Savart involves one integral, but it's a vector integral with a cross product inside.

That can be quite tricky to evaluate.

Finding dollar first involves solving for Escalers, ISA separately.

Each of these requires solving a scalar integral like Decade, JXR, DVD.

These three scalar integrals are often mathematically simpler to handle than the single vector integral of Biot -Savart.

So trade one hard vector integral for three easier scalar integrals.

That's the trade -off.

And for complex geometries, the dollar route is frequently the more manageable path.

Plus, as we saw with the solenoid, OLS seems to carry more fundamental information, especially important in relativity and quantum mechanics.

But dollar is what we measure directly via forces, but a dollar might be closer to the underlying reality.

What a journey.

Okay, so wrapping up.

We needed the vector potential dollar because magnetic fields are divergence -less, nobletod b equals other.

This means a dollar must be the curl of something, and that something is other.

But dollar isn't unique.

We have gauge freedom.

We usually pick the Coulomb gauge, nobletod a, for simplicity.

Which then leads to an equation for epsilon a, epsilon c2, that looks just like Poisson's equation for the scalar potential file.

Allowing us to calculate eight dollars using integrals analogous to electrostatics, often component by component, which can be easier than using Biot -Savart directly.

And the example showed it works, but more importantly, the solenoid revealed something profound.

Dollar can be non -zero even where the force field dollar is zero.

Exactly.

It suggests a dollar isn't just a mathematical trick, it reflects deeper physics, like the influence of enclosed flux.

It really positions eight dollars as the central quantity in magnetostatics, perhaps even more primary than a dollar itself.

Which leaves us with a really provocative thought, doesn't it?

If dollar can exist and have measurable effects, like an Aronoff -Bohm, in regions where dollar is zero, is a dollar the real field, and the dollar is just one of its manifestations.

That's the question that physicists continue to explore.

What is the fundamental nature of these fields?

The vector potential dollar certainly forces us to think beyond just the force field dollar.

Something for you to definitely ponder.