Faraday's Law: How Electromagnetic Induction Generates Electricity

Push a bar magnet into a coil of wire connected to nothing but itself, and — with no battery anywhere in sight — a current flows. Pull the magnet back out, and current flows again, in the opposite direction. Hold the magnet perfectly still inside the coil, no matter how strong it is, and the current vanishes completely. This single, strange-sounding demonstration is electromagnetic induction, and it is the working principle behind every electric generator, transformer, and induction cooktop on Earth.

This page is three linked interactive simulators in one. Swing a bar magnet through a coil and watch the induced EMF trace out its signature waveform. Spin a coil inside a field and watch the same law generate a clean AC sine wave. Then toggle a magnet's pole and direction and watch Lenz's law pick the current's direction for you, every time.

What Is Faraday's Law of Electromagnetic Induction?

Faraday's law states that a changing magnetic flux through a coil induces an electromotive force (EMF) around that coil:

$\varepsilon = -N \frac{d\Phi}{dt}$

where $\Phi$ is the magnetic flux through one turn of the coil, $N$ is the number of turns, and $\varepsilon$ is the induced EMF. The crucial word is changing — a magnet sitting motionless next to a coil produces a perfectly constant flux, and $d\Phi/dt = 0$ means no EMF at all, no matter how powerful the magnet is. Motion (or any other change in flux) is not optional; it is the entire cause of the effect.

The minus sign is Lenz's law, folded directly into the equation: the induced current always flows in the direction that opposes the change causing it — if flux is increasing, the induced current creates its own magnetic field to fight the increase; if flux is decreasing, the induced current tries to prop it back up. This is really just energy conservation in disguise: if the induced current instead reinforced the change, the effect would run away and generate energy from nothing.

Why Does Motion Matter? The Coffee-Grounds Way to Picture Flux

Picture magnetic field lines as grains of coffee scattered across a table, and the coil as a wire loop you can lower down flat onto the table or lift back up. Flux is simply a count of how many grains fall inside the loop's boundary. Lower the loop flat onto a dense patch and the count is high; lift it into empty air and the count is zero. Nothing dramatic happens while the loop sits still at a fixed height — the count doesn't change, so nothing is induced. It is only the act of moving the loop toward or away from the dense patch that changes the count, and it's that rate of change the equation cares about.

A moving bar magnet through a coil works the same way: the magnet's own field lines are densest close to its poles, and sliding it past the coil continuously changes how many of those lines pass through the loop. See how magnets produce that field pattern in the first place if you need the underlying picture.

Interactive Faraday's Law Simulator

Watch the magnet swing back and forth through the coil. The lower panel plots the induced EMF over one full swing, and if you follow it closely you'll notice it touches zero four times, not two. Two of those zeros happen when the magnet is centred in the coil, where flux is at its peak and its instantaneous rate of change is momentarily zero — the same idea as a ball at the top of its arc having zero vertical speed. The other two happen at the far ends of the swing, where the magnet itself is momentarily at rest — no motion, no changing flux, no EMF, even though the magnet is off-centre and flux is far from its peak there. Faraday's law only cares about the product of how fast flux changes with position and how fast the magnet is moving; either factor alone hitting zero is enough to silence the EMF. Drag the Swing period slider shorter to make the magnet move faster — the peak EMF grows even though nothing about the magnet or coil itself changed, because faster motion means faster flux change.

This simulation uses a simplified, illustrative model of how flux depends on the magnet's position — smooth and bell-shaped, peaking when the magnet is centred — rather than solving the full 3D field of a real magnetic dipole through a real coil. The resulting EMF waveform (zero at both the flux peak and the swing's turning points, alternating sign in between) matches how a real oscillating magnet-and-coil demonstration behaves; the numbers are illustrative, not laboratory-calibrated.

What Factors Increase the Induced EMF?

Faraday's law, $\varepsilon = -N\, d\Phi/dt$, shows every lever a student or engineer has for increasing an induced EMF:

• More turns ($N$) — each turn adds its own induced EMF, and they add up in series. Doubling the turns doubles the EMF for the same changing flux.
• Faster change ($d\Phi/dt$ larger) — moving the magnet faster, or building a stronger magnet, both increase the rate of flux change and therefore the EMF. This is why bicycle dynamos and hand-crank generators need to be spun quickly to produce useful power.
• Stronger field or larger coil area — both increase the peak flux $\Phi$ itself, giving a bigger range for it to change over.

How Does an Electric Generator Actually Work?

Every wall socket, every power line, and every wind turbine ultimately relies on the same trick you just saw with a sliding magnet, running continuously instead of twice per swing. Spin a coil of wire inside a magnetic field instead of sliding a magnet past it, and the flux through the coil changes constantly and smoothly — not because anything is translating back and forth, but because the coil's orientation relative to the field keeps changing.

With the coil's normal at angle $\theta = \omega t$ from the field direction, the flux through it is $\Phi(t) = BA\cos(\omega t)$. Differentiating and applying Faraday's law the same way as before:

$\varepsilon(t) = -N\frac{d\Phi}{dt} = NBA\omega\sin(\omega t)$

The result is a sine wave — alternating current, generated directly by rotation. Peak EMF $\varepsilon_0 = NBA\omega$ sets the ceiling: spin faster, use a stronger field, add more turns, or build a bigger coil, and the peak grows. This is the same law as the magnet-through-coil demo above, applied to a different kind of motion — rotational rather than translational — and it's the actual mechanism inside every generator and alternator that produces grid electricity, from hydroelectric turbines to bicycle dynamos. See how a coil's rotational motion connects to circular motion more generally.

Watch how the amber normal arrow lines up with the blue field arrows exactly when the EMF curve crosses zero, and points straight up or down (perpendicular to the field) when EMF peaks — flux and EMF stay 90° out of phase with each other, always. Spin the coil faster with the angular-speed slider and the curve compresses in time and grows taller at once: faster rotation shortens the period and increases $d\Phi/dt$ simultaneously.

How Do You Find the Direction of the Induced Current? (Lenz's Law)

Faraday's law gives the size of the induced EMF; Lenz's law gives its direction, and it's built directly into that minus sign in $\varepsilon = -N\,d\Phi/dt$. The rule in words: the induced current always flows in the direction that opposes the change causing it.

1. Determine whether the flux through the coil is increasing or decreasing.
2. If it's increasing, the induced current fights back — it flows to create a field that opposes the growing flux.
3. If it's decreasing, the induced current props it up — it flows to reinforce the field in the same direction.
4. Use the right-hand rule on that opposing (or reinforcing) field to read off the current's direction around the loop.

Toggle the magnet's pole and its direction of travel below, and watch both the induced current direction and the resulting force update:

Notice something that holds across all four combinations: the force on the magnet always opposes its own motion, regardless of which pole leads. Approach with either pole and the coil pushes back; withdraw with either pole and the coil pulls you back. You always have to do work to change the flux, and that work is the electrical energy delivered to the circuit — energy conservation, applied to magnetism. If the coil ever helped instead of resisted, a single push could spin a generator forever for free; nature doesn't allow it.

Worked Examples for Physics Exams

Example 1: Average EMF from a changing flux

A 200-turn coil experiences the flux through each turn change from 0.002 Wb to 0.008 Wb over 0.5 s. Find the average induced EMF.

$\varepsilon = N \dfrac{\Delta\Phi}{\Delta t} = 200 \times \dfrac{0.008 - 0.002}{0.5} = 200 \times 0.012 = 2.4$ V (magnitude — the sign depends on the chosen flux direction and Lenz's law).

Example 2: Finding the induced current

The coil in Example 1 has a resistance of 12 Ω. Find the induced current, treating the coil like a simple resistor driven by its own induced EMF (an application of Ohm's law — see our Ohm's law calculator for the general V = IR relationship).

$I = \varepsilon / R = 2.4 / 12 = 0.2$ A.

Example 3: Direction by Lenz's law

A bar magnet's north pole approaches a coil head-on. State the direction of the induced current using Lenz's law.

The approaching north pole increases the flux through the coil pointing away from the magnet. By Lenz's law, the induced current must oppose that increase — it flows in whichever direction makes the coil's near face into an induced north pole, since two north poles repel and resist the magnet's approach. If the magnet were instead being pulled away, the flux would be decreasing, and the induced current would reverse to make the near face a south pole, attracting the magnet back and again resisting the change.

Example 4: Peak EMF from a rotating generator coil

A generator coil has $N = 50$ turns, area $A = 0.08$ m², and rotates in a $B = 0.4$ T field at $\omega = 8$ rad/s. Find the peak EMF.

$\varepsilon_0 = NBA\omega = 50 \times 0.4 \times 0.08 \times 8 = 12.8$ V. Verify in the simulator: these are the default settings — the status readout shows the same peak.