Faraday's law
E = −N · ΔΦΔt
| Quantity | Symbol | Unit |
|---|---|---|
| Induced EMF | E | V |
| Number of turns | N | - |
| Flux | Φ | Wb |
| Time | t | s |
Elec Class
learn electricity hands-on
Lesson · chapter 11 of 15
When the magnetic flux through a coil changes, a voltage appears across it: this is induction, with a value of E = N × ΔΦ / Δt (Faraday's law), and the induced current always opposes the change that creates it (Lenz's law). A coil also reacts to changes in its own current: this is self-induction.
The result is worked out before your eyes, line by line, just like on the board.
1I'm looking for
2I enter the known values
Formula used E = N · ΔΦΔt
3The result
Induced EMF E30 V
If the flux through 500 turns changes by {ΔΦ} in {Δt}, the induced EMF is 30 V.
The definition, the key points to remember and everyday examples.
Faraday: any change of magnetic flux through a circuit induces an EMF E = −N·ΔΦ/Δt. Lenz: the induced current opposes the cause producing it (minus sign). A conductor cutting field lines at speed v gets E = B·l·v.
Self-induction: a coil opposes changes of its own current, E = −L·ΔI/Δt; L is the inductance, in henries. It stores W = ½ L I². When an inductive circuit is opened, the overvoltage strikes an arc: it is suppressed by a diode or an RC network.
A coil behaves like a loaded bike: hard to get moving, hard to stop dead. Inductance L plays the role of mass and current the role of speed; the energy ½ L I² even looks like the kinetic energy ½ m v². Braking suddenly takes a huge force, just as suddenly cutting the current in a coil creates a huge voltage. Where it falls short: the picture describes self-induction, not induction by an external magnet.
| Quantity | Symbol | Unit |
|---|---|---|
| Induced EMF | E | V |
| Number of turns | N | - |
| Flux | Φ | Wb |
| Time | t | s |
| Quantity | Symbol | Unit |
|---|---|---|
| Magnetic flux density | B | T |
| Length | l | m |
| Speed | v | m/s |
| Quantity | Symbol | Unit |
|---|---|---|
| Inductance | L | H |
| Current | I | A |
| Quantity | Symbol | Unit |
|---|---|---|
| Energy | W | J |
| Quantity | Symbol | Unit |
|---|---|---|
| Time constant | τ | s |
Four steps to solve a problem, then an example solved line by line.
The flux from outside (magnet, nearby coil), a conductor moving in the field, or the current in the coil itself.
Φ = B × S, then ΔΦ = Φ final − Φ initial; convert cm² to m² and ms to s.
E = N × ΔΦ / Δt for a changing flux, E = B × l × v for a moving conductor, E = L × ΔI / Δt for self-induction.
If the flux increases, the induced current creates an opposing field; if it decreases, a field in the same direction, to try to keep it going.
The flux through a 200-turn coil drops from 2 mWb to 0 in 10 ms: what induced EMF? A coil L = 0.5 H carrying 2 A: what energy does it store, and what voltage appears if the current is cut in 1 ms?
E = N · ΔΦ / Δt = 200 × 2 × 10⁻³ / 10 × 10⁻³ = 40 V (Lenz's minus sign only gives the direction: the induced current opposes the disappearance of the flux).
W = ½ · L · I² = 0.5 × 0.5 × 4 = 1 J.
E = L · ΔI / Δt = 0.5 × 2 / 10⁻³ = 1 000 V: hence the arc when an inductive circuit is opened, and the freewheeling diode that suppresses it.
The induced current flows in a direction such that its effects oppose the cause that produced it. If you bring the North pole of a magnet towards a coil, the coil presents a North pole to it and pushes it away: that's the − sign in Faraday's law.
E = B × l × v, when the conductor, the field and the velocity are mutually perpendicular. A 20 cm bar sliding at 5 m/s in a 0.5 T field: E = 0.5 × 0.2 × 5 = 0.5 V.
It's its tendency to oppose changes in its own current, written L and measured in henries (H). A 0.1 H coil whose current changes by 1 A in 10 ms produces E = 0.1 × 1 / 0.01 = 10 V.
About 5 × τ, with τ = L / R. A 0.2 H coil in series with 10 Ω has τ = 0.02 s: the current reaches 63 % of its final value after 20 ms and practically 100 % after 100 ms.
They are induced currents that circulate in loops inside a solid metal part placed in a changing flux. They heat it up: useful in an induction hob or a truck's retarder brake, a nuisance in a transformer, hence its core made of thin insulated sheets.
Most hobs are designed for a ferromagnetic base (cast iron, steel) that concentrates the field and heats strongly. Aluminum, non-magnetic and a very good conductor, heats little, and the hob often won't even start; a steel adapter disc solves the problem.
The coil opposes the sudden loss of its current: E = L·ΔI/Δt becomes huge when Δt is very short, and the air breaks down between the contacts. A freewheeling diode gives the current a path to die out gently.
Induction: the changing flux comes from outside (a magnet, another coil), as in a transformer or an alternator. Self-induction: the coil reacts to the change of its own current, hence its inductance L.
From the simplest to the most complete, each problem comes with a step-by-step solution.
Connect a 500 or 1,000 turn coil to a center-zero galvanometer, then move a bar magnet towards it, hold it still and move it away, slowly then quickly: students see that motion is needed, that the direction reverses with the direction of motion and that the deflection grows with speed.
Using Elec Class in class