Lecture notes · Physics: Electricity & Magnetism · Lecture 14

Maxwell's Equations and Electromagnetic Waves I

Yale PHYS 201 Fundamentals of Physics II, lecture 14, explained in plain English

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Part 1 · 0:00–10:00

This part sets up Maxwell's four equations and starts the path to waves by using a vibrating string and Newton's laws.

Key ideas

Magnetic field lines have no start or end, because there are no magnetic charges. Whatever goes into a closed surface must come back out. 2:10
Electric field lines do start and end on charges. Charges make the lines and absorb them, so the net flux out of a volume depends on the charge inside. 2:32
A changing magnetic field creates an electric field. A changing electric field or an electric current can create a magnetic field. 2:52
A wave equation can be found from Newton's laws alone. Apply F = ma to a tiny piece of a tight string, where the net sideways force comes from the string bending. 6:00
For small angles, the sideways force depends on the difference in slope at the two ends of the piece. Because we look at one fixed moment in time, the slope is written as a partial derivative. 9:14

Words to know

Tension: The pulling force along a string or rope, pulling on both of its ends.
Flux: The net count of field lines passing out through a surface. Positive means more lines leave than enter.
Partial derivative: A slope or rate of change measured in one direction while everything else is held fixed, such as a fixed moment in time.
Small angle approximation: When an angle is very small, measured in radians, you can treat sine and tangent as the angle itself and drop the tiny corrections.

Example

Think of a guitar string pulled tight between two posts. If you pluck it, it bends, and the string pulls back toward its straight shape because each side pulls at a slightly different angle. That pull back is what makes the string wave.

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Part 2 · 10:00–20:00

This part shows how a vibrating string leads to the wave equation, and why any shape that depends only on x minus vt travels to the right at speed v.

Key ideas

A derivative tells you how much something changes over a tiny step. If you subtract a function at two nearby points, the difference is approximately the derivative times the step size. 10:46
Applying Newton's law to a small piece of string gives the wave equation. The force from the string's tension equals the piece's mass times its acceleration, and this leads to d2y/dx2 minus (1/v2) times d2y/dt2 equals 0. 11:57
The symbol v equals the square root of T over mu. It is the speed of waves on the string, and it depends on the tension T and the mass per unit length mu. 12:45
Any function of the combination x minus vt is a solution to the wave equation, no matter what shape it has. Such a shape keeps its profile and slides to the right at speed v. 14:17
The most general solution is any function of x minus vt plus any function of x plus vt. Waves going right and waves going left can be added together, because the equation is linear. 19:24

Words to know

derivative: A number that tells you how fast something is changing at a point, like how steep a hill is at one spot.
wave equation: A rule that describes how a wave moves. Any shape that fits this rule can travel along the string without changing its form.
chain rule: A method for finding the rate of change of a function that is built inside another function, by working through each layer one step at a time.
linear equation: An equation where you can add two solutions together and get another solution, so waves can be stacked on top of each other.

Example

Flick one end of a long rope up and down, and you make a bump. The bump travels along the rope while keeping its shape, so it is the same bump moved a little further each second.

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Part 3 · 20:00–30:00

This part explains a shortcut for checking whether a field solution obeys Maxwell's equations everywhere: check tiny pieces, then add them up.

Key ideas

To test a field solution, you would have to check the equations for every possible surface and loop, which looks impossible. The lecturer says there is a quicker way. 23:28
If the equations hold for a tiny loop, tiny surface, or tiny volume, they hold for big ones too. The same is true for any arbitrary loop. 24:00
Two neighboring shapes glued together give a surface integral equal to the sum of their separate integrals. The shared face counts twice with opposite outward directions, so it cancels. 26:15
You can build any large shape from tiny pieces. If the integral is zero on every piece, it is zero on the whole shape. If each piece holds a charge, the total is the sum of those charges. 27:30
Line integrals work the same way. Two loops joined along a shared edge give a total equal to the sum of their separate line integrals. 29:32

Words to know

vacuum: An empty space with no matter, no charges, and no current flowing.
surface integral: A way of adding up a field's effect across a surface. You split the surface into small patches, multiply each patch's area by the field pointing through it, and add the results.
infinitesimal: So tiny that it is almost zero in size, but still a real small piece of the whole.
line integral: A way of adding up a field along a path or loop. You add the field's push along each tiny step of the path.

Example

Think of building a big wall from small bricks. If every brick is solid, the whole wall is solid, and when you glue two bricks together, the shared face is hidden inside and does not add to the outside. Checking tiny pieces of a field works the same way.

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Part 4 · 30:00–40:00

The lecturer checks that Maxwell's equations hold by testing tiny pieces, then tries a simple wave pattern and shows it passes a cube test.

Key ideas

If a Maxwell equation is true for a tiny square, it is also true for any bigger region, because the bigger region is made of many tiny squares whose inner edges cancel out. 30:58
A loop floating in 3D can be covered by a dome-like surface made of tiny tiles. The inside edges cancel, so only the rim of the surface is left. 31:44
To prove the rule for every loop, it is enough to check tiny flat loops in three directions: the xy, yz, and zx planes. 32:35
The lecturer makes a simple guess: the electric field points along z, the magnetic field points along x, and both change only with y and t. This is called a plane wave. 33:50
For this guess, the electric field is the same on the top and bottom faces of a small cube. The flux going in one side cancels the flux going out the other, so the total is zero, as the rule requires. 39:27

Words to know

infinitesimal: extremely tiny, so small that it is almost zero in size.
flux: how much of a field passes through a surface, like how much water flows through a hoop held in a stream.
plane wave: a wave where the field has the same value everywhere on a flat sheet, and the sheets are stacked one after another.
superposition: adding two solutions together to get a new solution, which works because the wave equation is linear.

Example

Think of a big wall built from tiny bricks. If every brick is level, the whole wall is level too. The lecturer uses the same idea: if a rule works on each tiny piece, it works on the whole shape.

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Part 5 · 40:00–50:00

This part checks whether a tiny cube and tiny loops of electric and magnetic fields obey two of Maxwell's equations, and finds a link between how Ez changes and how Bx changes over time.

Key ideas

Flux is the amount of field passing through a surface. On a tiny cube, the flux through each pair of opposite faces cancels or is zero, so the total flux of E is zero. 40:00
A field that runs parallel to a face, or is perpendicular to the face's area vector, gives no flux through that face. Opposite faces with the same field value and opposite area vectors cancel. 40:21
The cancellation only works because the field does not change across the cube in the direction being checked. If the field were stronger on one face than the other, the two faces would not cancel. 42:02
For a tiny loop, the line integral adds up the field along each edge. Edges that run opposite ways cancel, and edges perpendicular to the field add nothing, so the total can be zero. 44:35
A loop in the x-z plane gives a real condition: the line integral of E around it comes out proportional to how Ez changes with y, times the loop's area. This is set equal to the rate of change of the magnetic flux. 49:47

Words to know

flux: How much of a field passes through a surface. Think of how much rain goes through a hoop held up in the rain.
area vector: An arrow that points straight out from a flat surface, with its length equal to the surface's size. Its direction is set by the right hand rule.
infinitesimal: So tiny that you can treat it as a small piece where the field is almost the same everywhere.
line integral: A running total of a field's push along a path, such as around a loop, where only the part of the field pointing along the path counts.

Example

Imagine holding a hoop in a steady wind. If the wind blows parallel to the hoop's plane, no air goes through it, so the flux is zero. If the wind blows straight through, the flux is large, and it matters whether the wind is stronger on one side than the other.

3 checked practice questions for this part are in the app.

Part 6 · 50:00–1:00:00

Working through a loop of fields shows that changing electric and magnetic fields can travel through empty space as a wave moving at the speed of light.

Key ideas

Drawing a loop and comparing the line integral of the electric field with the surface integral of the magnetic flux gives the equation dEz/dy = -dBx/dt. Both sides depend on the loop's area. 50:53
If the fields follow the two rules from the loops, every Maxwell equation is satisfied. The rules are that the y-change of one field matches the time-change of the other. 53:40
Combining the two rules gives the wave equation, which describes something oscillating over space. Here the thing oscillating is the electric field itself, not a material like a string. 54:23
The wave speed comes out to 1 divided by the square root of μ0ε0, which is about 3×10^8 meters per second. That is the speed of light, which is why light is understood to be an electromagnetic wave. 55:32
The fields keep passing energy back and forth between electric and magnetic forms. You cannot destroy one without creating the other, so the wave sustains itself even with no charges present. 58:52

Words to know

line integral: A total added up along a path, such as all the field strength going around the edge of a loop.
surface integral: A total added up over an area, such as all the field passing through a flat surface.
flux: How much of a field passes through a surface, like how much water flows through a net.
wave equation: A rule describing how something ripples through space or along a string over time.

Example

The lesson compares a wave in these fields to an LC circuit. A charged capacitor pushes current through an inductor, and that current keeps flowing until the capacitor is charged the opposite way, so the energy swings back and forth. The fields do the same thing across all of space.

3 checked practice questions for this part are in the app.

Part 7 · 1:00:00–1:10:03

The professor finishes the math showing that the electric and magnetic fields form a wave that travels at the speed of light, and he describes what that wave looks like.

Key ideas

The professor plugs a wave guess into two Maxwell equations. Each equation must hold on its own, not just when they are combined into the wave equation. 1:00:48
The two equations give conditions on the sizes of the electric and magnetic waves. Together they force the wave speed to equal the speed of light, c. 1:03:12
When the wave speed equals c, the wave depends only on y minus ct, so the wave shape moves forward at the speed of light. 1:03:34
In the wave, the electric field and the magnetic field are at right angles to each other, and the wave travels in the direction of E crossed with B. The electric field is c times bigger than the magnetic field. 1:06:29
For slow charges like electrons in a radio antenna, the electric force is much bigger than the magnetic force, so the electric field does most of the pushing. 1:08:33

Words to know

Maxwell's equations: A set of equations that describe how electric and magnetic fields behave and how they relate to each other.
Amplitude: The biggest size a wave reaches, measured from the middle line up to the peak.
Wave equation: A rule that describes how a wave moves through space and time.
Propagation: The direction in which a wave travels.

Example

Think of a rope you flick up and down. The shape moves along the rope even though each bit of rope only moves up and down. In the same way, the electric and magnetic fields wiggle in place while the wave pattern moves forward at the speed of light.

3 checked practice questions for this part are in the app.

← Lecture 11: Lenz's and Faraday's Laws

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. Open Yale Courses · CC BY-NC-SA 3.0 · not affiliated with or endorsed by Yale. These notes are shared under the same license.

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