Lecture notes · Physics: Electricity & Magnetism · Lecture 11

Lenz's and Faraday's Laws

Yale PHYS 201 Fundamentals of Physics II, lecture 11, explained in plain English

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Part 1 · 0:00–10:00

This part explains how a changing magnetic field or a moving loop creates an electric push (voltage) in a loop of wire, and which way the resulting current flows.

Key ideas

Faraday's law says the electromotive force, which is the total push on a unit charge around a closed loop, equals the negative rate of change of the magnetic flux through that loop. 0:00
Flux is the amount of magnetic field passing through a loop. It can change because the field changes over time, because the loop moves through a field that varies in space, or both. 0:56
A changing magnetic field creates an electric field that is not made by electric charges. Unlike an electrostatic field, its line integral around a loop is not zero. 5:19
The minus sign, known as Lenz's law, means the induced current creates its own magnetic field that fights the change in flux you are causing. 5:41
When you drag a wire loop through a steady field, the motion creates the push, and the person pulling the loop supplies the energy that becomes electrical energy in the bulb. 9:37

Words to know

electromotive force: The total push that drives charges around a closed loop of wire. It is the work done on each unit of charge going around the loop.
flux: How much magnetic field passes through a loop or surface. Think of it as the total number of field lines going through an opening.
non-conservative: A field whose push around a closed path does not add up to zero. Going all the way around a loop can gain or lose energy.
Lenz's law: The rule that an induced current always flows in the direction that makes its own magnetic field oppose the change that caused it.

Example

Imagine pushing a bar magnet toward a loop of wire. The loop reacts as if it does not want more magnetic field passing through it, so a current starts up that creates a field pushing back against the magnet. It is like a door that pushes back when you try to close it too fast.

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

Part 2 · 10:00–20:00

This part explains how a changing magnetic field creates a real electric field that can push charges, and how a betatron uses that effect to speed up particles.

Key ideas

A changing magnetic field creates a real electric field, even in a loop of wire that is not moving. This is the main result from Faraday's equation. 10:00
A loop only gives a non-zero line integral if it crosses the region where the electric field exists. A loop that sits fully inside or fully outside that region gives zero. 11:00
When computing emf, you can ignore the part of the force from the current moving along the wire, because it is perpendicular to the wire and does no work going around the loop. 12:59
In a cyclotron, the orbit frequency stays the same as the particle speeds up, as long as the particle is not moving too fast. At high speeds, relativistic effects break this rule. 16:01
A betatron uses a changing magnetic field to create a circular electric field. That field pushes the particle so it speeds up while staying on a fixed-radius circle, even at relativistic speeds. 18:11

Words to know

flux: How much magnetic field passes through an area. Think of it as the total amount of field going through a hoop.
line integral: Adding up a field's push or strength along a path, step by step, all the way around a loop.
emf: Electromotive force. It is the push that drives current around a loop, measured as energy per unit charge.
betatron: A machine that speeds up charged particles in a circle using a changing magnetic field to create an electric field that pushes them.

Example

Imagine a fan blowing air in a closed room. Changing the fan speed changes the air flow, and the air pushes things around the room. In the same way, a changing magnetic field pushes charges around a loop by creating an electric field.

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

Part 3 · 20:00–30:00

This part explains how a changing magnetic field speeds up a charged particle in a circle (the betatron), why the field must be just right to keep it there, and how spinning a coil in a magnet makes electricity.

Key ideas

A magnetic field that changes over time creates a circling electric field. That field pushes a charged particle around a circle and makes it speed up. 22:28
Speeding up is not enough. A magnetic force must also pull the particle toward the center, and that pull has to grow as the particle goes faster. 21:23
To keep the particle on the same circle, the magnetic field at that circle must be exactly half of the average field inside it. 22:05
A charged particle moving in a tight circle at very high speed gives off energy as light. Eventually it loses energy faster than the push can supply, so the betatron fails. 24:48
When a coil spins in a magnet, the magnetic flux through it keeps changing. That change produces an electric push (emf), and the induced current pushes back against the change. 27:26

Words to know

flux: How much magnetic field passes through an area, like how much wind goes through a window.
emf: Short for electromotive force. It is the electric push that can drive current around a loop, like a battery's push.
Coulomb field: The electric field made by electric charges themselves. It pushes back on other charges that try to move.
synchrotron: A later kind of particle accelerator that uses other methods to deal with the energy lost when particles go around in circles.

Example

Swing a ball on a string in a circle. The string must keep pulling the ball toward your hand, or the ball flies off in a straight line. Faster swinging means you need a stronger pull, just as a faster particle needs a stronger magnetic pull to stay in its circle.

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

Part 4 · 30:00–40:00

This part explains how a spinning coil in a magnetic field makes voltage, who pays for the power it makes, and how a changing current in one coil can create voltage in a nearby coil.

Key ideas

Inside a wire, the magnetic push on moving charges is canceled by an electric push from charges that pile up at the ends. So the charges inside the wire feel almost no net force and can move freely. 30:37
When you spin a coil in a magnetic field, the two open ends show a voltage equal to the emf. Because the voltage keeps changing with time, this is how most household power supplies are made. 33:00
If nothing is connected, the spinning coil takes no work from you. Once you close the circuit with a load, current flows and a torque pushes back on the coil, so you must apply force to keep it turning. 35:13
The mechanical power you put in equals the electrical power lost in the resistor, which is I squared times R. The energy you spend turning the crank is the energy the load uses. 36:31
A changing current in one coil changes the magnetic flux through a second coil nearby, and that creates an emf in the second coil. This is the first effect Faraday noticed. 38:33

Words to know

emf: Short for electromotive force. It is the push that drives charges around a circuit, and it works like a voltage.
torque: A twisting force that turns something around an axis, like the force you feel when turning a crank.
flux: A way of counting how much magnetic field passes through a loop of wire. When flux changes, it creates an emf.
inductance: The tendency of a coil of wire to create a voltage when the current or magnetic field around it changes.

Example

Think of a ski lift. You are pulled up the hill by the lift, and then you ride back down because of gravity, with the lift's push exactly matching gravity while you are on it. Inside the wire, the magnetic push and the electric push cancel out in a similar way, so the charges are carried along the wire.

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

Part 5 · 40:00–50:00

This part explains why we can still use voltage in circuits when magnetic fields change over time, and introduces mutual inductance, which links a changing current in one coil to a voltage in a second coil.

Key ideas

Faraday found that a changing magnetic field can produce a current, but a steady field cannot. This is the basic reason circuits can get a voltage from magnetism. 40:00
Inside a wire, the induced electric field is cancelled by an electrostatic field built up by charges. That is why the electric field inside a wire stays at zero. 41:37
The changing magnetic field is hidden inside the coil. Outside the coil, the electric field is electrostatic, so a voltage can be defined there and used in circuits. 42:43
If a coil loops around more than once, the voltage is multiplied by the number of turns, because each loop adds its own voltage in series. 46:08
Mutual inductance tells you how much magnetic flux one coil produces in a second coil for each unit of current in the first coil. 47:17

Words to know

electrostatic field: An electric field made by charges that are sitting still or nearly still, like the field that builds up on the ends of a wire.
flux: The total amount of magnetic field passing through a loop or surface, found by adding up the field over the area.
emf: Electromotive force. It is the push that drives current around a circuit, measured as a voltage.
mutual inductance: A number that shows how strongly one coil's current creates a magnetic flux in a second coil, so a changing current in one coil produces a voltage in the other.

Example

Think of a water wheel that is only turned when the water level is changing. A steady flow does nothing, but a change in flow turns the wheel. In the same way, a steady magnetic field produces no voltage, but a changing one does.

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

Part 6 · 50:00–1:00:00

This part explains how a changing current in one coil can create a voltage in another coil, how transformers use this, and why a coil (an inductor) pushes back against changes in current.

Key ideas

Inductance is measured in henries. It tells you how much magnetic flux one coil produces per unit of current, and the same flux can pass through a second coil. 50:26
In a transformer, the voltage ratio between two coils equals the ratio of their numbers of turns, because they share the same flux. 51:10
A transformer does not make energy. The power drawn from the output must be supplied by the source, so it works more like a lever that trades distance for force. 52:40
A coil that carries a changing current makes its own flux, which sets up a voltage (called back emf) that opposes the change. 55:07
In a circuit with a battery, a resistor, and an inductor, the voltage going around the loop adds up to zero, with the inductor's drop equal to L times the rate of change of current. 58:50

Words to know

henry: The unit used to measure inductance. Common sizes are millihenries and microhenries.
transformer: A device with two coils wrapped around the same core that changes one voltage into another. It only works with changing current, such as AC.
self inductance: How much magnetic flux a coil makes through itself when current flows through it.
electromotive force: The push that drives charges around a circuit, sometimes called emf. Here it is the voltage created by changing flux.

Example

Think of a lever with a long arm and a short arm. A small push on the long arm can lift a heavy object on the short arm, but you have to move the long end much farther. A transformer works the same way: it can change the voltage up or down, but it cannot give you extra energy.

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

Part 7 · 1:00:00–1:08:36

This part explains that an inductor stores energy in its magnetic field, works out how big that field's energy is, and shows why the current in an inductor cannot jump instantly.

Key ideas

An inductor is a coil of wire that fights only a change in current, not the current itself. A resistor fights any current. 1:05:46
Building up current in an inductor takes work, and that work is stored as energy equal to one half times L times I squared. 1:01:52
For a simple solenoid, the self inductance is L = μ0 times n times N times A, where n is turns per unit length, N is the total number of turns, and A is the cross-sectional area. 1:03:47
A magnetic field holds energy spread through space. The energy density is B squared divided by 2μ0, which is energy per unit volume. 1:04:53
The current in an inductor can never jump instantly, because a jump would need an infinite rate of change and so an infinite voltage. Capacitor charge works the same way. 1:06:56

Words to know

inductor: A coil of wire that resists changes in the current flowing through it.
solenoid: A long coil of wire wrapped in loops, which makes a magnetic field inside when current flows through it.
flux linkage: The total magnetic flux passing through all the loops of a coil, added up loop by loop.
energy density: How much energy is stored in each unit of space, such as each cubic meter.

Example

Think of a heavy cart on a track. Getting it moving from rest takes effort, and that effort is stored as motion energy. An inductor works like this with current: it resists any change in its flow, and building up the current takes work that gets stored in its magnetic field.

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

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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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