Lecture notes · Calculus II · Lecture 33

Maxwell's Equations

MIT 18.02 Multivariable Calculus, lecture 33, explained in plain English

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Part 1 · 0:00–10:00

This part explains what the curl of a velocity field and the curl of a force field mean, and why a force that comes from a potential has zero curl.

Key ideas

Curl measures rotation. For a velocity field, curl gives twice the angular velocity, which is how fast something spins and which way it spins around. 1:16
Torque is the rotating effect of a force. It is the cross product of a position vector and a force, and it tells you how hard a force tries to swing an object around a point. 4:37
The curl of a force field tells you how much torque that field would put on a small solid placed in it, which shows whether the solid would start to spin. 7:41
If a force field has zero curl, it cannot make a still object start spinning. Friction could still slow down an object that is already spinning, but the field itself does not cause that. 8:26
Forces that come from a potential have zero curl. This ties back to energy: such forces never give you energy for free. 9:58

Words to know

curl: An operation on a vector field that measures how much the field makes things rotate or spin.
angular velocity: How fast something is spinning, along with the direction of the axis it spins around.
torque: The twisting or turning effect of a force about a point. It is a push that tries to rotate something.
conservative: A force is conservative when it comes from a potential, so the work it does depends only on the start and end positions, and energy is conserved.

Example

Picture holding a heavy ball at the end of a stick. The pull of gravity tries to swing the ball around the point where the stick meets your hand, and you must push back to keep it still. That swinging effect is torque, and the lesson uses this same picture to explain how a force can make things rotate.

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Part 2 · 10:00–20:00

This part introduces Maxwell's equations, which describe how electric and magnetic fields behave, and explains the first of them, the Gauss-Coulomb law, which links electric fields to electric charge.

Key ideas

The lecture first covers spin. Gravity alone cannot make a perfectly solid body start or stop spinning, so the Earth's spin was not caused by gravity. Friction and tidal effects in deformable bodies can line up rotations over time, as with the Moon always showing one face to Earth. 10:23
Maxwell's equations describe how electric and magnetic fields behave and how charges and their motion create them. They explain how electric devices work and how light and other electromagnetic waves travel. 12:35
An electric field is a set of arrows that tells you the push a charged particle would feel. The force equals the charge times the electric field. 14:06
The divergence of the electric field depends only on electric charge. In a region with no charge, the divergence is zero. 16:37
If you add up the electric field leaving any closed surface, the result depends only on the total charge inside that surface, divided by a constant. This is easier to picture than the differential equation and helps in studying capacitors. 18:05

Words to know

curl: A measure of how much a field makes things rotate or swirl at a point. If the curl is zero, the field causes no rotation.
divergence: A measure of how much a field spreads out or starts from a point. It tells you whether there are sources (like charge) there.
electric field: A field of arrows in space that shows the force a charged particle would feel if placed at that spot.
flux: The total amount of a field flowing out through a surface. Think of it as a count of how much of the field passes through a boundary.

Example

Imagine a closed box with some charge sitting inside. The lecture says the total field leaving the box's surface depends only on how much charge is inside, not on where it sits. A box with no charge inside has no net field leaving it.

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

Part 3 · 20:00–28:39

This part explains how a changing magnetic field can create an electric field and voltage, and how that idea makes transformers work.

Key ideas

Faraday's law says the curl of the electric field is linked to how fast the magnetic field changes over time. Curl is a measure of how a field swirls around a point. 21:07
Stokes' theorem connects a loop of wire to the surface it bounds. The voltage around the loop equals the flux (the total amount of field passing through that surface). 22:35
If the magnetic field through a wire loop is changing, a voltage appears in the wire, even with no battery. A transformer uses this effect. 23:18
Changing current in the first coil makes a changing magnetic field, which creates voltage in the second coil. This is how a transformer passes power between its coils. 24:03
The curl of the magnetic field is caused by moving charged particles, measured by current density J. This is the fourth Maxwell equation the lecture mentions. 26:58

Words to know

curl: A way of measuring how a field swirls or rotates around a point.
Stokes' theorem: A rule that links the total swirl or push around a closed loop to what passes through the surface inside that loop.
flux: The total amount of a field that passes through a surface.
transformer: A device that uses a changing magnetic field to pass electrical energy from one coil of wire to another.

Example

Think of a spinning fan blowing air through a ring: the faster the air flow changes, the more it pushes on anything in its path. In the same way, a changing magnetic field pushes on the electrons in a nearby wire and creates a voltage, even when nothing is connected to a battery.

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

← Lecture 31: Stokes' Theorem

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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