Lecture notes · Physics: Electricity & Magnetism · Lecture 10

Ampere's Law

Yale PHYS 201 Fundamentals of Physics II, lecture 10, explained in plain English

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Part 1 · 0:00–10:00

Ampere's law gives a shortcut for finding magnetic fields by adding up the currents that pass through a closed loop.

Key ideas

Ampere's law says that if you add up the magnetic field all the way around a closed loop, the result equals a constant (μ0) times the total current passing through the loop. Currents outside the loop do not count. 0:00
Currents are counted with a sign. Currents coming out of the page are positive and currents going in are negative. If they cancel to zero, the total is zero, even though currents are flowing. 0:50
The answer does not depend on which surface you use, as long as its edge is the same loop. Any current that enters one surface must leave through the others. 3:17
Symmetry makes some problems much easier. For a long straight wire, the field circles around the wire, and its strength depends only on the current and how far you are from the wire. 7:17
The next problem to study is the solenoid, which is a coil of wire wrapped many times around a tube. The goal is to find the field strength inside and outside the coil. 8:46

Words to know

line integral: A way of adding up a quantity, such as the magnetic field, all the way along a path or loop, taking into account the direction at each point.
current density: How much electric current flows through each unit of area, such as each square meter.
boundary: The edge or rim of a surface. For example, the edge of a soap film is its boundary.
solenoid: A coil of wire wrapped many times around a tube, which creates a magnetic field when current runs through it.

Example

Think of a wire frame dipped in soap. The frame is the loop, and the soap film is a surface with that loop as its edge. Whether the film is flat or bulges out, it still has the same edge, so the answer you get does not change.

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

Part 2 · 10:00–20:00

This part shows how Ampere's law gives the magnetic field inside a long coil of wire called a solenoid, and why the answer depends only on how tightly the wire is wound and how much current flows.

Key ideas

For a very long solenoid, the magnetic field outside is zero. The field lines only run along the length of the coil, so they have nowhere to go outside. 12:21
The field inside the solenoid is the same strength everywhere, and it points along the coil's length. The lesson uses a loop argument to show this. 13:14
When you apply Ampere's law, you can't just count the current in one wire. You have to count how many turns of wire pass through your loop, because each turn adds its current. 15:42
The loop is a made-up tool, so the final answer can't depend on its length. That is why the length cancels out. 17:08
The main result is boxed: the field inside a solenoid equals a constant times the turns per unit length times the current. The speaker says he uses this result all the time. 17:52

Words to know

solenoid: A coil of wire wrapped around a tube, so that current running through it creates a magnetic field.
Amperian loop: An imaginary loop you draw in space to help you use Ampere's law. It is not a real wire.
flux: The total amount of magnetic field passing through an area, like the amount of water flowing through a pipe.
turns per unit length: How many times the wire wraps around for each unit of length, such as each meter of the coil.

Example

Think of a spring made of wire. If you wrap the wire tightly, with more loops in each inch, the spring holds more of the effect of the current inside it. The result depends on how tightly it is wound and how much current flows, not on how long a measuring path you imagine around it.

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

Part 3 · 20:00–30:00

This part uses Ampere's law to find the magnetic field inside a doughnut-shaped coil, shows that a huge doughnut looks like a straight solenoid, and begins the story of why the old equations had to change.

Key ideas

To find the magnetic field inside a doughnut-shaped coil, you draw a circular loop, called an Amperian loop, and count the wire turns it passes through. The field is mu-zero times the current times the number of turns, divided by the circle's length (2 pi r). 21:35
The field is strongest near the inner edge of the doughnut and gets weaker toward the outer edge, because the same total current is spread over a bigger circle. Outside the doughnut the field is zero, because a loop there cuts equal amounts of incoming and outgoing current. 21:56
If the doughnut is enormous, it looks like a straight tube from close up. Its field then matches the straight solenoid formula: mu-zero times the number of turns per unit length times the current. 23:12
The equations for steady electric and magnetic fields work only when charges and currents do not change over time. New experiments showed they must be changed for situations where things change, and that is why Maxwell's equations are needed. 28:01
When you drag a wire loop through a magnetic field, the light bulb in the loop lights up even though no battery is attached. This is a change caused by motion in a magnetic field, and the lesson is about to explain it. 29:48

Words to know

Amperian loop: An imaginary closed path you draw around a current. You use it to work out how strong the magnetic field is along that path.
solenoid: A coil of wire wrapped many times around a tube. It makes a fairly uniform magnetic field inside.
magnetostatics: The study of magnetic fields that stay the same over time, from steady currents.
Maxwell's equations: The full set of rules that describe electric and magnetic fields, including cases where things change over time.

Example

Think of a garden hose bent into a big ring with water flowing through it. Near the inside of the ring the water is squeezed into a tighter curve, while farther out it spreads over a wider path. The magnetic field of the doughnut coil behaves the same way: it is stronger on the inside and weaker on the outside.

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

Part 4 · 30:00–40:00

A loop of wire moving through a magnetic field makes a push that lights a bulb, and the energy comes from whoever is pulling the loop.

Key ideas

The bulb glows even though there is no battery. Each charge that goes once around the loop gets some work done on it, so something must be pushing the charges. 30:21
That push is the magnetic force on moving charges. It acts only on the part of the wire that moves across the field, so the emf is the field strength times the speed times the wire's width. 31:30
The magnetic field seems to do work even though it should do none. The charges move along the wire and also with the loop, so the real velocity is tilted, and the field does work on one part and takes work away on the other. 33:36
The field does not create energy. It moves energy from the person pulling the loop to the electrons, which are too small for anyone to push by hand. 36:06
The energy books balance. The power used up in the resistor equals the power you supply by pulling, so the setup works like a generator. 38:37

Words to know

emf: Short for electromotive force. It is the total push, per unit of charge, that drives charges around a loop, and it is measured as work done per charge.
resistor: A part in a circuit that slows the flow of charge and turns energy into heat or light, like the filament inside a bulb.
v x B force: The magnetic push on a moving charge. It points at right angles to both the charge's motion and the magnetic field.
generator: A device that turns mechanical work, like pulling or spinning, into electrical energy.

Example

A bicycle dynamo lights the front lamp while you pedal. Your legs supply the energy, the spinning magnet and coil move it into the wires, and the lamp glows only as long as you keep pedaling.

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

Part 5 · 40:00–50:00

When a magnetic field changes through a loop, it creates an electric field that pushes charges around the loop, and the size of that push is set by how fast the magnetic flux changes.

Key ideas

Once the loop is fully inside the magnetic field, the bulb stops glowing. The push on one side of the loop cancels the push on the other side, so the charges have no net push to go around. 41:04
Relativity says the loop and magnet can be viewed in either frame, but the bulb must glow in both. So a changing magnetic field must create an electric field, even when no charges are nearby. 42:36
This electric field is new. It does not come from charges, and its line integral is not zero, so it is not conservative. Work is done on charges as they go around the loop. 45:27
The exact rule cannot be worked out from logic. It comes from experiment: the electromotive force around a loop equals minus the rate of change of the magnetic flux through the loop. 47:47
The surface you choose for the flux does not matter. Magnetic field lines never stop or start, so whatever goes through one surface must go through any other surface with the same edge. 49:21

Words to know

Magnetic flux: A measure of how much magnetic field passes through a surface. You find it by adding up the field across the surface.
Electromotive force (emf): The work done on each unit of charge as it travels once around a loop. It is what pushes current around a circuit.
Line integral: A running total of a field's push along a path, added up piece by piece around a loop. If the total is zero, the field does no net work on a charge going around.
Conservative: A field is conservative if moving a charge around a closed loop gives zero net work. The new electric field here is not conservative.

Example

Think of a fixed loop of wire and a magnet being carried past it. Whether you say the magnet moves or the loop moves, the relative motion is the same, so the bulb must glow the same way. The only sensible explanation is that the changing magnetic field creates an electric field that pushes the charges around the loop.

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

Part 6 · 50:00–1:00:00

This part explains Lenz's law, which says a changing magnetic flux creates a current that works against that change, and it applies this idea to a loop moving through a magnetic field.

Key ideas

When the magnetic flux through a circuit changes, it induces an emf (a push that can drive a current). If the circuit allows it, the current flows to fight that change. 50:17
The current's own magnetic field points in whatever direction works to cancel the change. If flux is going up, the induced current makes flux that pushes it back down. 51:13
Lenz's law opposes the change in the magnetic field, not the external field itself. This is what the minus sign in the law means. 51:37
For a loop pulled through a fixed field, the flux is the field times the width times the length of loop still inside. Its rate of change matches the emf found earlier. 54:11
The emf equals the rate of change of flux. Once the loop is fully inside the field, the flux stops changing, so the current stops. 56:48

Words to know

flux: How much magnetic field passes through a given area, like how much wind goes through a window.
emf: Electromotive force: a push that can make electric current flow in a circuit, measured in volts.
Lenz's law: The rule that an induced current always flows in the direction that works against the change that caused it.
boundary: The edge that surrounds a surface, like the rim of a cup or the outline of a flat shape.

Example

Think of a bike rider trying to speed up a spinning wheel that is already moving. The wheel's own resistance pushes back against the change, the way the induced current pushes back against a change in flux. Like the wheel, the current does not stop the change completely, but it fights it.

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

Part 7 · 1:00:00–1:13:13

This part shows that a changing magnetic field creates a circulating electric field, which is the rule known as Faraday's law.

Key ideas

The magnetic flux through a loop can change for two reasons: the magnetic field itself can change over time, or the loop can move and sweep across the field. 1:01:57
When the loop moves, the extra change in flux comes from the edge of the loop sweeping out a small strip, and this adds a term that depends on the velocity v and the magnetic field B. 1:06:26
Faraday's law says the line integral of the electric field E around a loop equals the negative rate of change of magnetic flux through that loop. A changing magnetic field produces an electric field that circulates around it. 1:09:10
This relation is a property of electric and magnetic fields themselves. It does not depend on whether there is a real wire or conducting loop present. 1:10:19

Words to know

flux: A measure of how much magnetic field passes through a surface or loop, like counting how many arrows pass through a hoop.
emf: Short for electromotive force, the push that drives electric charge around a loop, measured as energy per unit charge.
contour: A closed path or outline in space that you walk around when doing a line integral.
Faraday's law: The rule that a changing magnetic field makes an electric field that goes around in circles, so its loop total is not zero.

Example

Think of a magnet being pushed toward a coil of wire. As the magnet moves, the amount of magnetic field through the coil changes, and that change pushes electricity around the coil. The wire does not have to be the cause; the changing field is what matters.

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

← Lecture 6: Capacitors · 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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