Electromagnetic Induction

In short: Electromagnetic induction is making a voltage (emf) by changing the magnetic flux through a coil: Faraday's law, $ε = - N \frac{ΔΦ}{Δ t}$ . Lenz's law says the induced current always opposes the change that made it. Generators and transformers work this way.

Magnetic flux

Magnetic flux $Φ$ ("phi") measures how much magnetic field passes through an area, like counting the field lines going through a loop.

$Φ = B A cos θ$

$B$ = field strength (T)
$A$ = area of the loop (m²)
$θ$ = angle between the field and the normal (a line at right angles to the loop's surface)
The unit is the weber (Wb). 1 Wb = 1 T m².

The flux is biggest when the field goes straight through the loop face-on ( $θ = 0°$ ). It is zero when the loop is edge-on to the field.

Faraday's law

Michael Faraday discovered that a voltage (an emf, electromotive force) is made in a coil whenever the flux through it changes. A steady flux does nothing.

$ε = - N \frac{ΔΦ}{Δ t}$

$N$ = number of turns in the coil
$ΔΦ$ = change in flux (Wb)
$Δ t$ = time taken (s)

Faster change, more turns or a bigger change all give a bigger emf. The minus sign is Lenz's law (below).

Ways to change the flux:

Move a magnet towards or away from a coil.
Change the current in a nearby electromagnet.
Rotate the coil in a field (this changes $θ$ ).
Change the area of the loop, for example by sliding a rod along rails.

Lenz's law

The induced current flows in the direction that opposes the change that caused it.

Push a magnet's north pole towards a coil: the coil's near end becomes a north pole to push it back.
Pull the north pole away: the near end becomes a south pole to pull it back.

Lenz's law is conservation of energy. If the induced current helped the magnet along instead, you would get electrical energy and extra motion from nothing. You always have to do work against the opposing force, and that work becomes the electrical energy.

Eddy currents

In a solid piece of metal, a changing flux makes swirling loops of current called eddy currents. By Lenz's law, they oppose the motion that causes them.

Magnetic braking: trains and roller coasters use magnets near metal fins. The eddy currents slow them smoothly, with no contact and no wear. The kinetic energy becomes heat in the metal.
Eddy currents are unwanted in transformer cores, because they waste energy as heat.

Transformers

A transformer changes the size of an AC voltage. It has two coils wound on a shared iron core:

AC in the primary coil makes a changing magnetic field in the core.
The core carries this changing flux to the secondary coil.
The changing flux induces an AC emf in the secondary coil.

There is no electrical connection between the coils. Only the changing flux links them, which is why a transformer does not work on steady DC.

For an ideal transformer (no energy lost):

$\frac{V _{p}}{V _{s}} = \frac{N _{p}}{N _{s}} V_{p} I_{p} = V_{s} I_{s}$

Step-up: more turns on the secondary. Voltage goes up, current goes down.
Step-down: fewer turns on the secondary. Voltage goes down, current goes up.

Electromagnetic Induction

Flux

Φ = BA cos θ
Weber, Wb
Max face-on, zero edge-on

Faraday's law

ε = −N ΔΦ / Δt
Only a changing flux
Faster, more turns: more emf

Lenz's law

Opposes the change
Conservation of energy

Eddy currents

Swirling currents in metal
Magnetic braking
Waste heat in cores

Transformers

Vp / Vs = Np / Ns
VpIp = VsIs
AC only

Year 11A rod sliding on rails

A metal rod of length $l$ slides at speed $v$ across a field $B$ , along two rails joined at one end. The loop's area grows by $l v$ every second, so:

$ε = \frac{ΔΦ}{Δ t} = B l v$

The charges in the rod are moving through a field, so each feels $q v B$ along the rod, which pushes them to one end. You can find which end becomes positive with the palm rule (thumb along the rod's motion). If a current flows, the motor effect then pushes back on the rod, against its motion: Lenz's law again.

Only the part of the field across the motion counts. A plane's wings sweep out area horizontally, so only the vertical part of Earth's field cuts them: $ε = B_{vertical} l v$ . With $B = 5.0 \times 1 0^{- 5}$ T at 30° to the vertical, $B_{vertical} = B cos 30°$ .

Which end charges up. For a bar moving through a field with nothing connected, use the palm rule on a positive charge (thumb along the motion, fingers along $B$ ): the palm points to the end that becomes positive. The other end becomes negative.

Changing shape or angle. Pulling a loop of wire from a square into a circle increases its area (a circle encloses the most area for a given length), so the flux changes and an emf appears. Turning a coil from face-on to edge-on changes the flux from $B A$ to 0.

Currents next to a wire. If the current in a nearby straight wire increases, the flux through a loop beside it increases. The induced current opposes this: it flows the opposite way to the wire's current on the near side, so the loop is pushed away.

Year 12Reading flux and emf graphs

Since $ε = - N \frac{ΔΦ}{Δ t}$ , the emf is $- N$ times the gradient of a flux against time graph.

Flux steady (flat line): emf is zero.
Flux rising steadily: emf is constant.
Flux changing fastest (steepest part): emf is largest.

For a coil rotating at steady speed, the flux is a cosine wave and the emf is a sine wave: the emf is zero when the flux is greatest, and greatest when the flux is zero (coil edge-on).

Real transformers are less than 100% efficient: $efficiency = \frac{V _{s} I _{s}}{V _{p} I _{p}} \times 100%$ . Losses come from:

Eddy currents in the core. Fix: build the core from thin insulated sheets (laminations).
Resistance of the coils heating them. Fix: thick copper wire.
Flux leakage, where some flux misses the secondary. Fix: wind both coils on one closed iron core, close together.

Each section of a graph. If a field rises steadily for 2 s, stays constant for 2 s, then falls to zero in 1 s, the emf is constant in the first part, zero in the middle, and larger (and reversed) in the last part, because the flux falls faster.

Magnets and pipes. A magnet falls slowly through a copper pipe because eddy currents oppose its motion, but it falls freely through a plastic pipe, which cannot carry current. Cutting slots in a metal plate breaks up the eddy current loops, so a slotted plate swings through a field with much less braking than a solid one.

Open circuits. An emf is induced whether or not the circuit is closed, but current flows only in a closed circuit. A magnet falling through a coil with its ends open feels no braking force.

Efficiency from data. For 11 000 V and 50 A in, the input power is 550 kW. At 95% efficiency the output is 522.5 kW, so at 415 V the output current is about 1260 A.

Recoil. When current is switched on in a solenoid next to a copper block, the eddy currents in the block repel the solenoid. If both sit on trolleys, momentum is conserved: they move apart, the lighter trolley faster.

Quick summary

Flux $Φ = B A cos θ$ , measured in webers.
A changing flux induces an emf: $ε = - N \frac{ΔΦ}{Δ t}$ (Faraday's law).
The induced current opposes the change that made it (Lenz's law). This is conservation of energy.
Eddy currents form in solid metal. They give magnetic braking, and waste energy in transformer cores.
Transformers need AC: $\frac{V _{p}}{V _{s}} = \frac{N _{p}}{N _{s}}$ and, if ideal, $V_{p} I_{p} = V_{s} I_{s}$ .

Quiz

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A magnet sits still inside a coil. The induced emf is:
1. A Zero
2. B Large and steady
3. C Large and changing
4. D Equal to the magnet's strength
Show answer
A: Zero
There is no change in flux, so there is no induced emf.
The unit of magnetic flux is the:
1. A Tesla
2. B Weber
3. C Volt
4. D Ampere
Show answer
B: Weber
Flux is measured in webers (Wb). 1 Wb = 1 T m².
A loop of area 0.020 m² is face-on to a 0.50 T field. The flux through it is:
1. A 25 Wb
2. B 0.040 Wb
3. C 0.010 Wb
4. D 0.52 Wb
Show answer
C: 0.010 Wb
Φ = BA = 0.50 × 0.020 = 0.010 Wb.

Magnetic flux

Faraday's law

Lenz's law

Eddy currents

Transformers

Quick summary

Quiz

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