Chapter 22: Gauss's Law

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Okay, so today we are diving deep into something that's, well, pretty fundamental in magnetism.

It's called Goss's Law,

and it's going to change how you look at electric fields, especially if you've ever felt like you're wrestling with like a million point charges and Coulomb's Law is just not cutting it.

You know what I mean?

Absolutely.

You know those Christmas lights you spent hours untangling?

Goss's Law, it's like finding that master switch for the whole string.

Suddenly,

everything makes sense.

Exactly.

And for anyone listening who really wants to get a solid grasp on this, you've come to the right place because today's all about Goss's Law, but not just the formulas, but like the deep insights that this thing reveals.

It's about making those connections between charge and fields that might have seemed kind of mysterious before.

You hit the nail on the head.

By the time we're done with this deep dive, you'll be able to look at the electric field on a closed surface, any closed surface, and you'll know just from that what's going on with the charge inside.

Pretty neat, right?

Seriously neat.

We're going to demystify something called electric flux, figure out how to calculate it, and then get to the heart of the matter, Goss's Law itself, the direct link between this flux and the charge trapped inside our surface.

And we won't stop there.

We're going to equip you with the tools to use Goss's Law to find those electric fields for some beautifully symmetrical charge distributions, spheres, cylinders, infinite planes, things that might have seemed super complicated before.

All right, so to get started, let's try to visualize this whole flux thing, right?

Imagine you've got a pipe and you're thinking about how much water flows through a certain section of that pipe every second.

That's a kind of flux, a flow.

Turns out we can think about electric fields in a similar way.

Electric flux, it's kind of like the flow of that field through a circus.

Kind of sets the stage for what's coming, right?

A perfect analogy.

Now, remember how we used to spend all that time figuring out

created by charges,

adding up contributions from each point charge?

Well, Goss's Law flips the script.

It asks, what if we know the electric field in a region?

What can that tell us about the charges hiding there?

Okay, let's imagine we've got a closed box, could be a real box, or just an imaginary one.

It's our surface, right?

And maybe there's some charge inside that box.

How could we figure out how to put charges in there without like opening the box?

Well, electric field is our clue.

Since charges make those fields, by checking the field outside the box, we can figure out what's going on inside.

If the field looks like it's being pushed away from the box, you know, like a positive charge would push it.

Well, then we can guess there's a net positive charge in there.

Figure 22 .1 in the book shows this with a little test charge, kind of like a probe.

So we're like electric detectives.

And you know what?

We don't even need a full 3D map of the field.

We can just focus on the field at the surface of the box.

If you take a look at figure 22 .2, you'll see something really important.

If there's positive charge inside, the field lines point outward through the surface.

More charge, stronger field, but the direction is still outward.

Negative charge.

Well, those lines point inward towards the negative charge.

Makes sense, right?

It's all about the direction.

This is where electric flux comes in.

Outward field lines, outward flux.

Think of it as the electric field flowing out of the box.

Inward field lines, inward flux.

So just from that, from the direction of the field, we can figure out there's more positive or negative charge inside.

So what happens if there's no net charge inside the box, like zero total charge?

Ah, good question.

Take a look at figure 22 .3.

Yeah.

The first part shows an empty box.

No charge, no field, no flux.

Makes sense.

But in the next part, we've got both a positive and a negative charge inside, but they're equal and opposite.

So they cancel each other out, right?

Exactly.

There's an electric field, but some of it's going into the box, some of it's going out.

The net effect.

Zero flux.

All balanced out.

And you know what?

Even if there's an electric field coming from outside the box, like in figure 22 .3C, it still works the same way.

If there's no charge inside the box, the flux in equals the flux out.

Zero net flux.

Okay.

So we figured out how the sign of the charge inside affects the flux.

But what about the amount of charge?

Let's look at figure Trum 2 .4.

In the first part, we've got a positive charge plus Q in a box.

In the second part, we double the charge to plus two Q.

Same box.

What happens?

The electric field gets stronger.

It does.

Twice as strong, in fact, at every point on the surface.

It's like going back to our water analogy.

We've doubled the flow rate.

This tells us something very important.

The magnitude of the flux is directly proportional to the magnitude of the charge inside.

More charge, more flux.

And here's something even interesting.

It doesn't matter how big the box is.

You got it.

Look at figure 22 .4C.

Same charge plus Q, but the box is twice as big.

Now we know the electric field gets weaker the further away you are from the charge.

Right.

The inverse square law.

Exactly.

So the field on the surface of the bigger box is weaker on average than on the smaller box.

But the bigger box also has a lot more surface area.

And guess what?

It balances out.

Perfectly.

Weaker field, more area,

same total flux.

So what do we learn?

The flux depends on the charge inside, not the box itself.

That's mind -blowing.

So the flow of the field really is tied to the charge, not to our imaginary box.

That's pretty profound, isn't it?

It is.

It hints at something fundamental.

Let's sum up what we know about Gauss's law so far.

First, positive charge inside, outward flux.

Negative charge, inward flux.

Second, charges outside our box, they don't affect the total flux through the surface.

And finally, the more charge we pack in, the greater the flux, no matter the size or shape of the box.

We're seeing a pattern here, right?

It's like the charge inside is linked to this

flow of the electric field through any surface we draw around it.

And Gauss's law turns this into a precise mathematical statement, a law of nature.

But how do we go from this intuitive idea of flux to actually calculating it?

Let's start simple.

Imagine a uniform electric field, nice and constant, passing through a flat surface.

Right, so for a uniform electric field E passing through a flat surface, area A, the electric flats we call EE,

is given by EE equals EA times the cosine of phi, where phi is the angle between the electric field and, get this, the normal to the surface.

The normal.

Yeah, think of it like a little arrow pointing straight out from the surface like a flagpole.

It tells you the surface's orientation relative to the field in prudent detail.

We can write this using the dot product from vector math as AE equals E dot A, where A is the vector area with magnitude A in direction given by that normal vector.

Right, the dot product.

Got it.

So if the surface is perpendicular to the field lines, phi is zero degrees, cosine of zero is one, and the flux is just EA maximum flux, right?

But if the surface is parallel to the field lines, phi is 90 degrees, cosine of 90 is zero, and boom, zero flux.

Figure 22 .6E shows this.

Really clear.

Absolutely.

Now, the units for flux, just so you know, are newton meters squared per coulomb.

And remember, when we're dealing with closed surfaces, those imaginary boxes, the normal vector always points outward.

Positive flux, outward flow, negative flux, inward flow.

Gotta keep track of the signs.

Right, outward positive makes sense.

But what if the field isn't uniform or the surface isn't flat?

Things get more complicated then, right?

They do.

But thankfully, calculus comes to the rescue.

Imagine chopping up the surface into tiny little pieces, each with its own area, DA, and its own normal vector.

The flux through each piece is DE equals E dot DA, which is the same as E times cosine phi times DA.

To get the total flux, we sum up all those tiny contributions with an integral.

So EE equals the integral of E dot DA, or the integral of E times cosine phi times DA over the whole surface.

That's a surface integral.

Kind of like measuring the wind on a landscape, right?

The wind speed and the direction the hill is facing are always changing.

So you got to consider each little patch of the hill separately.

Precisely.

And in our case, we're considering the flow of the electric field through each little patch of the surface.

Now, if the field happens to be uniform and the surface is flat,

this fancy integral just simplifies back down to our earlier formula, EA cos phi.

It's good to see how things connect, right?

And have you seen that cool bio application in the text?

He talks about basking sharks, how the flux of water through their huge mouths is like electric flux.

Big flux means the shark gets a lot of plumped in.

Just like a big electric flux might mean there's a lot of charge enclosed.

Cool stuff, right?

It is.

Okay, so we have a way to actually calculate the flux.

What does the book do with this next?

Well, example 22 .2 shows you how to find the flux through a cube in a uniform electric field.

It's a good example because you calculate the flux through each face of the cube separately and then add them up.

And guess what?

When there's no charge inside the cube, the total flux always comes out to be zero.

It's like what we saw earlier.

The flux going in cancels out the flux going out.

Always comes back to that, huh?

What about that example with the sphere?

Ah, yes.

Example 22 .3.

This one calculates the flux through a sphere centered on a point charge.

A classic example.

Because of that spherical symmetry, the calculation is pretty straightforward.

The electric field from a point charge, it always outward.

Same strength everywhere on the sphere.

And what do we get?

The flux equals q over epsilon naught, where q is the charge and epsilon naught is our old friend, the electric constant.

And here's the kicker.

The radius of the sphere doesn't matter.

It cancels out in the math.

Wait, so no matter how big the sphere is, as long as the charge is in the middle, the flux is the same.

Exactly.

And if you look at figure 22 .11, you can see why.

They show two spheres, one small, one big, both centered on the charge.

Every field line that goes through the small sphere also goes through the big one.

Same number of lines, same flux.

That's just wild.

Right.

It means the flux is truly connected to the charge itself, not to how we choose to surround it with a surface.

And that test your understanding question after this section really drives that point home.

It gives you different flat surfaces of various angles to a uniform field and asks you to figure out the flux.

It's all about that dot product and understanding the angle between the and the normal vector.

Okay, so we've got electric flux down pat.

We can calculate it.

We see these hints of a deeper connection between charge and flux.

Now, are we ready for the main event?

Gauss's law itself.

We are.

This is where it all comes together.

Gauss's law, named after Carl Friedrich Gauss, brilliant guy.

It's a fundamental law in electromagnetism.

It gives us another way to express the relationship between charge and field.

Think of it like, you know, Newton's laws of motion and the work energy theorem, different ways to look at the same physics.

Right.

Different perspectives, same underlying truth.

So what does Gauss say?

Well, remember how we kept seeing that the total flux through any closed surface is proportional to the total charge inside?

Gauss's law turns that into a precise equation.

Let's start with a point charge, Q, right in the middle of a sphere, radius R.

Like in that example we just talked about.

Exactly.

We already know the

is Q over epsilon naught.

Now here's the key point.

That equation, equation 22 .6 in the book,

it doesn't depend on R, the radius of the sphere.

It only cares about the charge and that fundamental constant.

Doesn't matter how big or small the sphere is, same flux.

You got it.

And figure 22 .11 shows it beautifully.

The flux is tied to the charge, not to the surface we draw around it.

I'm starting to see why this is so important.

It goes beyond just like calculating things.

It's a statement about the nature of charge and field.

Absolutely.

And guess what?

It gets even better.

That relationship holds true even if the surface isn't a sphere.

Like imagine the most bizarre, lumpy, asymmetrical surface you can think of as long as it's closed.

The total flux is still Q over epsilon naught.

Figure 22 .12 in the book shows an example.

They even use a fancy math trick to prove it.

But the takeaway is clear.

The shape of the container doesn't change the total flow if the source of that flow, the charge, stays the same.

So no matter how weird the surface, same flux for the same charge.

That's powerful.

It is.

And you always have to remember that the surface has to be closed.

Like that circle on the integral sign, equation 22 .7, that little circle means we're talking about a complete closed surface.

Yeah.

A box, a sphere, whatever.

But it has to enclose a volume.

And the little da's in that integral, those tiny areas, they always point outward from the enclosed volume.

Gotta keep those directions straight.

Right.

Outward is positive, inward is negative.

We've got that down.

What if there's no charge inside our surface at all?

What does Gauss say then?

Well, if there's no charge inside, the total flux has got to be zero.

That's what equation 22 .8 says.

Yeah.

And it makes sense, right?

If there's no source of the field inside the surface, any field lines that go in must also come out.

They can't just end there like bangling in space.

They gotta start on a positive charge and end on a negative charge.

Figure 22 .13 illustrates this really well.

Okay.

So no charge, no net flux.

What if we've got like a bunch of charges inside our surface?

The general case.

So let's say we have a bunch of charges, Q1, Q2, Q3, and so on, all inside our surface.

The total charge, we call it Q enclosed, is just the sum of all those individual charges with their plus or minus signs, of course.

And Gauss' law says for any closed surface, the total flux equals that Q enclosed divided by epsilon naught.

It's equation 22 .8, the heart of Gauss' law.

So we just add up all the charges inside, divide by epsilon naught, and boom, that's our flux.

Yep, that's it.

But here's something really important to remember.

When we're talking about the electric field in that integral, we're talking about the total field from all charges everywhere inside and outside the surface.

But only the charges inside contribute to the net flux.

That's a key point.

Charges outside can affect the field, but they don't change the total flux through the surface, right?

Exactly.

They can mess with the field locally, but the overall balance of flux in and flux out, that's all about what's inside.

Figures 22 .14 and 22 .14 show some great examples.

They really help you see how the sign of the charge inside determines whether the flux is outward or inward.

And that tests your understanding question after section.

That's a good one.

It gives you a bunch of charges and a bunch of surfaces and asks you to rank them by flux, really drives home the point that it's all about the charge inside.

Absolutely.

Gauss' law gives us this beautiful connection between charge and field.

And now we're going to see how we can use it to actually solve some problem, to calculate things.

And this is where that whole symmetry thing comes in, right?

Symmetry, like how symmetrical the charge distribution is.

You got it.

Gauss' law works best, like really shines, when the charges are in a nice symmetrical way.

Think spheres, cylinders, infinite planes.

In those cases, because of the symmetry, we can often guess the direction and even the strength of the field.

And that makes the math way easier.

Okay.

So symmetry is our friend.

Anything else that makes us easier?

Yes.

There's this really important thing about conductors, materials where charges can move freely.

When you put some extra charge on a conductor and let it settle down, all that

really?

Why is that?

Well, think about it.

If there were a field inside the conductor, the charges would feel a force and start moving.

But we're talking about electrostatic equilibrium.

Everything's at rest.

So the field inside must be zero.

Otherwise the charges wouldn't stay put.

Okay.

That makes sense.

No field inside means no charge inside, right?

You got it.

If we draw a Gaussian surface inside the conductor, like in figure 22 .17, the flux through that surface has to be zero because the field is zero everywhere on it.

And Gauss's law says that if the flux is zero, the enclosed charge must also be zero.

So no charge inside.

And this is true even if the conductor has a hole in it, like a hollow shell.

As long as there's no charge in the hole, the inside surface of the conductor has no charge.

So all the action happens on the surface.

That simplifies things.

It does.

And problem -solving strategy 22 .1 lays it all out for you how to use Gauss's law step by step, the first step, figure out the symmetry, spherical symmetry, use a sphere as your Gaussian surface, cylindrical symmetry, use a cylinder,

planar symmetry, use a little cylinder, they call it pillbox, with its ends parallel to the plane.

You choose your weapon wisely, right?

What then?

Then you calculate the flux through your surface.

And because you've chosen a good surface based on the symmetry, this part is usually pretty easy.

The field might be perpendicular to the surface, might be parallel, might even be zero in some places.

And once you know the flux, you just plug it into Gauss's law and bam, you can solve for whatever you're looking for.

Could be the field strength, could be the amount of charge.

Pretty slick.

It is.

So let's try it out.

Example 22 .5, the charged conducting sphere.

We put a charge q on a sphere, radius r.

We want to find the electric field.

So spherical symmetry, we use a spherical Gaussian surface, right?

Exactly.

Concentric with a sphere.

Let's start with a point outside the sphere.

So our Gaussian surface has a radius r bigger than r.

That surface encloses all the charge, right?

Right.

Now because of the symmetry, the field at every point on our Gaussian surface must be pointing radially outward, same strength everywhere.

Makes the calculation super easy.

And we get this.

The field strength is q divided by 4 pi epsilon naught r squared.

Wait a minute, that's the same as the field from a point charge.

You got it.

From the outside, a charged sphere acts exactly like all the charges concentrated at its center.

Amazing, right?

Right.

Doesn't matter how the charge is spread out on the surface, it's like it's all bunched up at the center.

Mind blown again.

What about a point inside the sphere?

Well, now our Gaussian surface is inside the conductor, right?

And we know the field inside a conductor is zero,

always.

So zero field, zero flux, zero enclosed charge.

It confirms what we said before.

All the charges on the surface.

And it's the same even if it's a hollow shell, as long as there's no charge inside the hole.

Okay, that all makes sense.

What about that line of charge example?

Ah, yes.

Example 22 .6.

Now we have a line of charge, infinitely long, really thin charge per unit length lambda.

We want to find the field.

This one's cylindrical, right?

So we use a cylindrical Gaussian surface.

You know it.

Coaxial with the line of charge, radius r, length del.

The field from the line charge, it points radially outward.

Strength depends only on the distance from the wire.

Now, the flux through the ends of our cylinder,

zero.

The field is parallel to the ends, no flow through them, so all the flux is through the curved side surface.

We crunch the numbers using Gauss's law, and what do we get?

The field strength is lambda divided by two pi epsilon not r.

And I bet that matches what we got using that complicated integral from before, right?

It does.

Gauss's law makes it way easier.

Symmetry is our friend.

Okay, now for example, 22 .7, an infinite sheet of charge, charge its density sigma uniform across the whole sheet.

We want the field.

This one's planar, right?

So we use that pillbox thing.

Yep.

Little cylinder, axis perpendicular to the sheet.

The field from an infinite sheet, it's perpendicular to the sheet, same strength everywhere, same direction.

Now, what about the flux through our pillbox?

The field is parallel to the side, so zero flux there.

All the flux is through the two ends.

And if the pillbox encloses an area A of the sheet, the enclosed charge is sigma times A.

We use Gauss's law, and boom.

The field strength is sigma divided by two epsilon not.

And it points perpendicular to the sheet.

Away from it, if sigma is positive.

Towards it, the sigma is negative.

Another result we got before, but in a much simpler way.

Makes you wonder why we even bothered with all those integrals before.

Well, they're still important for situations that aren't so symmetrical.

But Gauss's law definitely makes life easier when we can use it.

Now, what about example 22 .8?

Two parallel plates, opposite charges, like a capacitor.

Ooh, that's a good one.

It is.

So we use these clever Gaussian surfaces that go from between the plates into the plates themselves, where we know the field is zero.

And we find something interesting.

The field between the plates is uniform, nice and constant.

And its strength is sigma divided by epsilon not.

Pointing from the positive plate to the negative plate.

And outside the plates, the field is basically zero.

It's like the fields from the two plates cancel each other out everywhere except between them.

It makes sense, right?

It's all about superposition, adding up the fields from each plate.

Okay, what about example 22 .9?

This one's a bit different.

It's a sphere again.

But it's uniformly charged throughout its volume, not just on the surface.

Good catch.

This is a uniformly charged insulating sphere.

Now, the field inside the sphere isn't zero anymore because the charges are spread throughout the volume.

We use a spherical Gaussian surface inside the sphere.

And this time, it only encloses a portion of the total charge.

We work it all out and find that the field inside the sphere is proportional to the distance from the center.

q r divided by 4 pi epsilon not r cubed, where q is the total charge and r is the sphere's radius.

So it gets stronger as you move further out from the center.

Exactly.

And outside the sphere, we get the same results as before.

q divided by 4 pi epsilon not squared, like all the charges at the center again.

Same as the conducting sphere.

Once you're outside.

Exactly.

So inside and outside, different behaviors.

And finally, example 22 .0.

This one's cool.

It shows you how to use Gauss's law backwards.

If you know the field outside a spherical charge distribution,

you can figure out the total charge inside.

Just set up a spherical Gaussian surface, calculate the flux, and solve for q enclosed.

So we can go both ways, from charge to field and from field to charge.

That's powerful.

And that test your understanding question reminds us that while Gauss's law is always true, it's most useful for calculating things when there's a lot of symmetry.

Absolutely.

Now let's dive a little deeper into what happens with charges on conductors.

We know they all go to the surface, but what about conductors with holes in them with cavities?

Like a hollow sphere.

Exactly.

So if there's no charge inside the cavity, and we draw a Gaussian surface inside the conductor around the cavity, the enclosed charge has to be zero because the field inside is zero.

And that means there's no charge on the inner surface of the cavity either.

It's like the conductor shields the cavity from any charge inside.

No charge on the inner surface.

Interesting.

What if we put a charged object inside the cavity?

Now things get interesting.

Imagine a little metal sphere, charge plus q, inside a hollow conducting shell, figure 22 .23c.

We draw our Gaussian surface inside the shell, enclosing the sphere in the cavity.

The enclosed charge has to be zero, right?

Because the field inside is zero.

Right.

So to cancel out that plus q inside, there must be a charge of in q on the inner surface of the cavity.

It's like the conductor is trying to neutralize the field from the charge inside.

It's like the conductor is fighting back.

Exactly.

And because the shell started out neutral, that not q on the inside has to be balanced by a plus q somewhere else.

And where does that go?

To the outer surface of the shell.

So a charge inside the cavity induces an equal and opposite charge on the inner surface and an equal charge of the same sign on the outer surface.

Wow, it's like pushing charges around.

What if the shell already had some charge on it before we put the sphere inside?

Same principle.

You still get that knee q induced on the inner surface to cancel out the charge inside, but now the total charge on the outer surface is the initial charge plus the plus q that was pushed out there.

So it adds up.

That's conceptual example 22 .11, right?

Yep.

And you know what?

This leads to some amazing things like Faraday's ice pail experiment.

Ever heard of that?

Rings a bell, but refresh my memory.

Faraday took a metal pail, initially uncharged, and lowered a charged ball into it, but without letting them touch.

Because of induction,

the ball induces charges on the inner and outer surfaces of the pail.

Like what we were just talking about.

Exactly.

Now when Faraday let the ball touch the inside of the pail, it became part of the inner surface.

And guess what?

The ball lost all its charge.

It got neutralized by the induced charge on the inside of the pail.

So it's like all the charge from the ball went into the pail?

Yep.

And that's exactly what Goss's law predicts.

Because the net charge enclosed by a surface inside the pail, including the ball, has to be zero.

It was a huge confirmation of Goss's law and Coulomb's law.

Pretty cool to see how this theoretical stuff actually plays out in experiments, right?

Absolutely.

And this whole idea of induction, it's the key to how those Van de Graaff generators work, you know, those things that make your hair stand on end.

They use a belt to carry charge to the inside of a hollow sphere.

And that charge goes straight to the outer surface.

You got it.

Because you can't stay on the inner surface.

Yeah.

And as more and more charge builds up on the outside, you get this crazy strong electric field.

So cool.

And this also explains electrostatic shielding, right?

Heritage cages.

Exactly.

If you make a box out of a conductor, any electric fields outside get canceled out inside the box.

It's because the free charges in the conductor move around until they create induced charges on the inner and outer surfaces that perfectly cancel out the external field inside.

That's why it's safe to be in a car during a lightning storm.

You got it.

Car acts like a Faraday cage.

Pretty handy, huh?

And the book mentions this cool bio -application, too, about how the charge distribution in a nerve cell is all about these same principles.

So this stuff is everywhere, from lightning to our own bodies.

What about that equation that connects the field just outside a conductor to the charge density on the surface?

Ah, yes.

E perpendicular equals sigma over epsilon -naught.

Equation 22 .N zero.

So E perpendicular is the part of the field that's pointing straight out from the surface, and sigma is the charge per unit area at that point.

You can derive this using a little pillbox -shaped Gaussian surface, one end just outside the conductor, one end just inside.

And the field inside is zero, right?

You know it.

So all the flux is through the end outside.

Crunch the numbers, and you get that equation.

It works for any shape conductor.

You can try it out for a plate or a sphere, see if it holds up.

Example 22 .N two uses it to figure out the charge on the earth.

Pretty wild.

And that box about lightning bolts, that connects it all to the earth's electric field.

Amazing, right.

And the last test your understanding question, it's all about how grounding a conductor neutralizes the outer surface, even if there's still charge trapped in a cavity inside.

Really makes you think about how these charges move around.

We covered a lot today, but it feels like everything's connected now.

It's like we unlocked a whole new level of understanding about electric fields and charge.

Totally agree.

We started with this intuitive idea of flux, this flow of the electric field.

And now we see how it all ties together with Gauss's law, this fundamental relationship between charge and field.

And it's not just theory, right?

It explains so many real world things, from shielding to how charge behaves on conductors.

For those listening, what were some of your aha moments today?

Was it seeing that the shape of the Gaussian surface doesn't matter, just the charge inside?

Or maybe realizing how much easier Gauss's law makes those calculations, especially when there's symmetry.

It's incredible how elegant this law is, isn't it?

It cuts through all the complexity and reveals the core relationship between charge and field.

But here's the thought.

If Gauss's law is so powerful for electrostatics, what about other areas of electromagnetism?

Are there similar laws waiting to be discovered, laws that could simplify our understanding of even more complex phenomena?

Now that's a question to ponder.

And while you're at it, think about this.

How does this idea of flux and a law like Gauss's law apply to magnetism?

Because after all, electricity and magnetism are two sides of the same coin, right?

What are the limits of Gauss's law in its electrostatic form?

Where does it break down?

And what new tools do we need for those situations?

And finally, can you now see the world around you differently, recognizing the role of electric fields and charge in everyday things, maybe even things you hadn't thought about before?

Keep those questions in mind as you continue your journey into the world of electromagnetism.

Thanks for joining us on this Deep Dive.