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In the early 1800s, Michael Faraday showed us how a changing magnetic field induces an electromotive force, or emf, resulting in an electric current. He also found that electric fields sometimes act like magnetic fields, and developed equations to calculate the forces exerted by both. In the mid-1800s, Scottish physicist James Maxwell thought something interesting was going on there, too. So he decided to assemble a set of equations that held true for all electromagnetic interactions. In this episode of Crash Course Physics, Shini talks to us about Maxwell's Equations and how important they are to our understanding of Physics.


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 Introduction (0:02)

By the 19th century, we were pretty sure there was something going on between electricity and magnetism.

First, in the early 1800's, Michael Faraday showed us how a changing magnetic field induces an electromotive force, or EMF, resulting in an electric current. He also found that electric fields sometimes act like magnetic fields, and developed equations to calculate the forces exerted by both.

In the mid-1800's, Scottish physicist, James Maxwell, thought something interesting was going on there too, so he decided to assemble a set of equations that held truth for all electromagnetic interactions. And he did a great job of it, compiling a set of equations as critical to physics as Newton's Laws of Motion.

While he was coming out with his all-encompassing equations, Maxwell predicted the existence of "electromagnetic waves". These are the waves that help you view the world around you, talk to your friends a thousand miles away, and heat up that midnight breakfast burrito that you've been craving.

So Maxwell's equations are, to put it simply, enlightening!

When James Maxwell created his dream-team of electromagnetic equations, he started with the ideas and the mathematics that other physicists had already established. But he put them together to create a new, complete understanding of electromagnetism.

 Maxwell's 1st Equation (1:22)

Maxwell's first equation is a form of Gauss's Law, which states that the electric flux through a closed surface is proportional to the total charge enclosed by that surface. A closed surface, in this case, is just a surface of any shape that completely encloses a charge, like a sphere, or any other 3-dimensional shape.

And electric flux is a measure of the electric field that passes through a given area.

Mathematically, this equation says that electric flux is the integral of the electric field over the area of the surface, which is equal to the enclosed charge divided by the permittivity of free space. 

I won't blame you if you don't remember, but the "permittivity of free space" is a constant proportionality that we've used before that relates electric charge to the physical effect of electric fields.

If the electric field is constant over the surface, then the left side of the equation becomes E times A.

 Maxwell's 2nd Equation (2:08)

Maxwell's second equation is also a form of Gauss's Law, only with magnetic flux instead of electric flux.

Magnetic sources are always dipoles, with a North and South pole, and magnetic field lines leave from the North pole and return back to the South.

When you're looking at how a magnetic field interacts with a surrounding surface, you see that the field lines that leave the surface out of the North pole must enter back through the surface to reach the South pole.

This results in the overall magnetic flux being zero.

If you compare this with Maxwell's first equation, you can tell that the format, with the integral of the magnetic field over the surface area, is the same. But here, the integral will always equal zero for a closed surface.

 Maxwell's 3rd Equation (2:45)

Now, Maxwell's third equation doesn't have anything to do with Gauss's Law, instead it's Faraday's Law, but slightly different than how you've seen it before. This is how you've seen it before:

If you remember, Faradays' Law of induction says that a changing magnetic field will induce the electromotive force in a loop of wire. Maxwell's version is a more general, simplifying Faraday's law to show the value of that induced EMF. 

It says that EMF is equal to the line integral of the electric field over a closed loop. So with this equation, you can see the connection between a changing magnetic flux, and an induced electric field.

 Maxwell's 4th Equation (3:17)

The fourth and final equation is one that Maxwell made the most tweaks to in order to make it applicable to all electromagnetic phenomena. Maxwell’s first and second equations were so similar, one for electric fields and one for magnetic fields, that he knew there must be an equation that complemented Faraday’s Law as well.

After all, if a changing magnetic flux produces an electric field, why is there not a complementary law saying the reverse?

Why can a changing electric flux result in an induced magnetic field? Here, Maxwell started with Ampere’s Law, which states that current thrill wire induces a magnetic field around a path surrounding the wire.

In order to make this equation work for all possible electromagnetic scenarios, Maxwell needed to consider a situation where a magnetic field is induced, but not from the flow of current.

For example, picture a charge capacitor that is connected to wires on both sides. As the capacitor charges, currents run through both wires, generating a changing magnetic flux, just like Ampere’s Law says.

But a changing magnetic flux is also induced between the two plates when there is no wire connecting the plates. However, there is a changing electric field and it acts like a wireless current.

In order to account for the magnetic flux generated by this wireless current, Maxwell added what is known as the displacement current into Ampere's law to complete the equation. Despite being called the displacement current, it isn’t really an electric current, it is just called that because it fits into the equation, just like an actual current does.

Mathematically, the displacement current is equal to the change in the capacitor’s charge over time, and that is equal to the change in electric flux over time multiplied by the permittivity of free space constant.

With the addition he made to Ampere’s law, Maxwell's equations were complete. And using his laws today, we can see a pattern in the production of electric and magnetic fields.

If a changing electric field is generated, then a magnetic field is induced, which results in a changing magnetic field that induces an electric field and the cycle continues.

 Electromagnetic Waves (5:00)

These oscillations are electromagnetic waves and they're made up of changing electric and magnetic fields that travel through space. Maxwell believed that we could reproduce electromagnetic waves in the laboratory. And he was right.

To prove it, let's look at an object that can produce electromagnetic waves and transmit them through the air: an antenna. 

This antenna is made up of two conductive rods, each attached to opposite ends of an A/C generator. When the antenna is operating, the two rods switch between being positively charged and negatively charged as the generator changes voltage. And when one rod is positively charged, the other is negatively charged, and vice versa. 

When electric current travels up into the top of the rod, you can use the Right Hand Rule, pointing your right thumb in the direction of the current to see how the magnetic field flows around the antenna.

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