Faraday's Law of Induction

In one sentence: A changing magnetic field through a loop of wire pushes a voltage into the wire, and that is how generators make electricity.

$\varepsilon = -N\, \frac{d\Phi_B}{dt}$

Read it out loud: “epsilon equals minus N times d Phi B over d t. The induced emf, epsilon, equals minus the number of turns N, times the rate of change of the magnetic flux, d Phi B over d t. Phi is the Greek letter phi, said fye.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Magnetic field $B$ and the tesla (T): A magnetic field is the influence that a magnet or a current has on the space around it. Its strength is measured in tesla (T), where one tesla is one newton per ampere per meter. Field lines are a way to draw the field: they are closer together where the field is stronger, and they point the way the field points.
Magnetic flux $\Phi_B$ and the weber (Wb): Magnetic flux, written $\Phi_B$ with the Greek letter phi, measures how much field passes through a loop. You get it by multiplying the field strength by the area that the field passes through. One weber (Wb) is one tesla times one square meter, so $1\ \text{Wb} = 1\ \text{T m}^2$ . Flux is like counting the field lines that thread through a hoop.
Area, square meters and powers of ten: Area is measured in square meters, $\text{m}^2$ , which is the area of a square 1 m on each side. A square centimeter is smaller. Since 1 m is 100 cm, 1 m $^2$ is $100 \times 100 = 10^{4}$ cm $^2$ , which is 10,000 square centimeters. So 100 cm $^2$ is $100 / 10^{4} = 0.01$ m $^2$ . A power of ten like $10^{4}$ is ten multiplied by itself four times.
Rate of change, $\frac{d\Phi_B}{dt}$ and $\Delta$: A rate of change says how fast something changes, like speed, which is distance divided by time. $\frac{d\Phi_B}{dt}$ is read ‘d phi B by d t’, and it means how fast the flux changes at each moment, in webers per second. The letter $d$ stands for a very small change. The triangle $\Delta$ , read ‘delta’, means a change over a time interval, so $\Delta\Phi_B / \Delta t$ is the average rate over that interval.
Electromotive force (emf) and the volt: Electromotive force, written emf and given the symbol $\varepsilon$ (epsilon), is a push that drives charges around a circuit. Its unit is the volt. Despite the old name, an emf is a voltage: the number of volts that a source can push charges with. A battery’s emf is close to the voltage printed on its label.
Normal to a surface and the cosine: The normal to a flat loop is a line that sticks straight out of its face, at right angles to it. The angle $\theta$ is measured between the magnetic field and that normal. The cosine of the angle, $\cos\theta$ , is 1 when the field runs straight through the face and 0 when the field runs flat across the face.

What each symbol means

Symbol	Meaning
$\varepsilon$	The induced emf, the voltage pushed around the loop by the changing flux, measured in volts (V). A sign of plus or minus tells you the direction of the push. The example’s induced emf is 5 V, the same voltage as a USB charging port.
$N$	The number of turns of wire in the coil, which is a count with no unit. Each turn is a loop that sees the same change in flux, so their voltages add together. A coil with 100 turns gives 100 times the emf of a single loop in the same change.
$\Phi_B$	The magnetic flux through one turn of the coil, measured in webers (Wb). One weber is one tesla times one square meter. The flux counts how much magnetic field passes through the loop’s area, and a changing flux is what induces the emf.
$t$	Time, measured in seconds (s). The emf depends on how quickly the flux changes, meaning how much the flux changes in each second. In the example the field grows over 0.1 s, which is one tenth of a second.
$B$	Magnetic field strength, measured in tesla (T). It is the field that passes through the loop. In the example it grows from 0 to 0.5 T, which is thousands of times stronger than the Earth’s magnetic field at its surface.
$A$	Area of the loop, measured in square meters (m $^2$ ). It is the flat area inside the loop, such as the face of a flat coil. The example area is 0.01 m $^2$ , which is 100 square centimeters, or a square 10 cm on each side.
$\theta$	The angle between the magnetic field and the normal to the loop, which is a line sticking straight out of the loop’s face. It is measured in degrees. At 0 degrees the field runs straight through the face and the flux is greatest. At 90 degrees the field runs flat across the face and the flux is zero.

The formula piece by piece

$\Phi_B = BA\cos\theta$: Start with the flux through the loop. Multiply the field by the area, then by the cosine of the angle. The cosine keeps only the part of the field that passes straight through the face of the loop.
$\frac{d\Phi_B}{dt}$: The rate of change of the flux, in webers per second. This is how fast the field lines passing through the loop are changing. The emf responds to this rate of change, and not to the flux itself.
$N\,\frac{d\Phi_B}{dt}$: Multiply the rate of change by the number of turns. Every turn sees the same change in flux, so their emfs add up. A coil with more turns gives more emf for the same change in flux.
$\varepsilon = -N\, \frac{d\Phi_B}{dt}$: The whole formula in one sentence: the induced voltage in a coil equals minus the number of turns times the rate at which the magnetic flux through each turn is changing.
$|\varepsilon| = N\,\frac{\Delta\Phi_B}{\Delta t}$: The average emf over a time interval, written as a size with the sign left out. Divide the change in flux by the time it took, then multiply by the number of turns. This is the form used in the worked example.

What the formula is doing, and why

The magnetic flux $\Phi_B$ counts how much magnetic field passes through a loop of area $A$ , measured in square meters (m $^2$ ). For a field $B$ , in tesla (T), at an angle $\theta$ to the loop’s normal, $\Phi_B = BA\cos\theta$ , measured in webers (Wb). Think of flux as the field lines that thread through the loop, like the water that passes through a hoop held in a stream.

Why only a change? A steady flux induces no emf, however strong the field is. The emf comes from the change: the formula says the voltage depends on how fast the flux changes, in webers per second. A field that grows quickly induces a bigger emf than one that grows slowly. A coil with $N$ turns multiplies the effect, because every turn sees the same change.

Why the minus sign? The minus sign is Lenz’s law. The induced current always flows in the direction that makes its own magnetic field oppose the change in flux that caused it. If the flux is growing, the induced current pushes back against the growth. This sign keeps energy conserved, because an induced effect that helped the change causing it would create energy from nothing.

Why it powers generators. Turning a coil in a magnetic field changes its angle $\theta$ , which changes the flux through the coil and produces an alternating emf. That is how generators make the AC of the grid. The same law explains transformers, where a changing current in one coil makes a changing flux that induces a voltage in the other coil.

When it applies. The law gives the emf around a closed loop of wire, and it works for any way the flux changes: a moving magnet, a moving coil, or a field that is switched on and off. For a flat coil in a uniform field, the flux is $BA\cos\theta$ . For the size of an average emf, drop the minus sign and use $|\varepsilon| = N\,\Delta\Phi_B/\Delta t$ .

An everyday picture

Think of a hoop held in a stream of water. The flux is how much water passes through the hoop. Tilt the hoop and less water gets through, which is the cosine in the formula. Change the flow or the tilt quickly, and the rate of change pushes a current around the hoop.

Worked example: Average emf from a field that grows in 0.1 s

A coil with $N = 100$ turns and an area of $A = 0.01$ m $^2$ , which is 100 cm $^2$ , is placed so that the magnetic field passes straight through its face. The field grows steadily from 0 to 0.5 T in 0.1 s. Find the size of the average induced emf.

Convert the area to square meters. One square meter is 10,000 square centimeters, since 1 m is 100 cm and $100 \times 100 = 10^{4}$ . So $100\ \text{cm}^2 = 100 / 10^{4}\ \text{m}^2 = 0.01\ \text{m}^2$ . The formula needs square meters, so this conversion comes first.
Find the angle factor. The field passes straight through the face, so it runs along the normal and $\theta = 0$ . Then $\cos 0^\circ = 1$ , and the cosine does not reduce the flux at all.
Find the flux at the end. $\Phi_B = BA\cos\theta = (0.5\ \text{T})(0.01\ \text{m}^2)(1) = 0.005\ \text{Wb}$ . Tesla times square meter gives webers, so the unit is right.
Find the change in flux and the rate of change. The flux starts at zero, so the change is $\Delta\Phi_B = 0.005 - 0 = 0.005$ Wb. Divide by the time, $\Delta t = 0.1$ s, to get $\Delta\Phi_B / \Delta t = 0.005 / 0.1 = 0.05$ Wb/s. The flux grows by 0.05 webers every second.
Multiply by the number of turns. $|\varepsilon| = N \times \frac{\Delta\Phi_B}{\Delta t} = 100 \times 0.05\ \text{Wb/s} = 5\ \text{V}$ . Each of the 100 turns sees the same change, so their voltages add up to 100 times the single-turn value.
Read the sign. The formula gives $\varepsilon = -5$ V. The minus sign is Lenz’s law: the induced current pushes in the direction that opposes the growing flux. Only the size, 5 V, is asked for in this example.
See how the time matters. If the same change of 0.005 Wb happened in 0.01 s instead, the rate would be ten times bigger, and the emf would be $100 \times 0.005 / 0.01 = 50$ V. A quicker change always gives a bigger induced voltage.

Answer: The average induced emf has a size of 5 V. Its sign is negative, so by Lenz’s law the induced current flows in the direction that opposes the growing flux through the coil. The size of 5 V is the same as the output voltage of a USB charging port.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Forgetting the number of turns. A 100 turn coil has 100 times the emf of a single loop in the same change.
Expecting a steady field to induce an emf. An emf needs a change in flux, so a constant field through a still coil induces nothing.
Using square centimeters without converting. 100 cm $^2$ is 0.01 m $^2$ , so using 100 would make the emf 10,000 times too big.
Ignoring the angle. If the coil is turned edge-on to the field, so the field runs flat across its face, $\cos\theta$ is zero and the flux is zero, however strong the field is.

Where you will meet it

Electric generators in power stations, where spinning coils in magnetic fields produce the alternating voltage of the grid.
Induction cooktops, where a changing magnetic field induces currents in the metal of the pan, which heat it.
Wireless phone chargers, which use a changing field in one coil to induce a current in a coil inside the phone.
Metal detectors, which sense the changing magnetic field from currents induced in nearby metal.

Questions people ask

Why does the emf depend on the rate of change and not on the field itself?

A steady field gives a steady flux, and a steady flux induces nothing. The emf comes from the change. A field that grows quickly induces more emf than one that grows slowly, even if both end at the same strength.

What is a weber?

It is the SI unit of magnetic flux. One weber is one tesla times one square meter, so $1\ \text{Wb} = 1\ \text{T m}^2$ . The flux at the end of the example is 0.005 Wb.

Why is there a minus sign in the formula?

The minus sign is Lenz’s law. The induced current always acts to oppose the change that causes it, which keeps energy conserved.

Transformer Turns Ratio: A transformer changes an AC voltage up or down by the same ratio as the number of wire loops on its two coils.
Magnetic Field of a Solenoid: A long coil of wire makes a strong, nearly even magnetic field along its middle, and the field gets stronger when more loops are packed into each meter of coil.
Inductor Voltage: A coil of wire makes a voltage across itself only while its current is changing, and the faster the current changes, the bigger that voltage is.
Magnetic Force on a Moving Charge: A magnetic field pushes a moving charge sideways, and the push is strongest when the charge moves straight across the field.