## Summary

In the SL Magentism course, you will have studied the motor effect, in which a current-carrying conductor or moving charge experiences a force in a magnetic field.

In the HL course, we move to electromagnetic induction. When a magnetic field is changed or moved relative to an electrical conductor, an EMF is induced. When the conductor is connected as part of a complete circuit, a current flows.

Faraday's law enables us to calculate the size of the induced voltage. First we will need to define a new concept - magnetic flux.

## Get the basics

### Electromagnetic induction

When a charge moves in a magnetic field, it experiences a force that is perpendicular to its velocity. This causes free charges to move in a circular path (NB: Centripetal force). The direction of this force can be determined using Fleming's left hand rule:

However, when constrained to a linear electrical conductor (i.e. a metal wire), charges cannot follow a circular path. If the wire continues to move in a straight line, electrons all electrons will experience a force in the same direction and move. Now that the electrons are not distributed evenly throughout the conductor, the wire has an electric field across it (like a battery). An EMF has been produced, that would cause a current to flow if the wire was connected to a resistor in a complete circuit.

### Magnetic flux

#### Area in the field

Magnetic flux is defined as the product of the magnetic flux density (NB: magnetic field strength) and the perpendicular area of the field.

\(\Phi=BA\cos\theta\)

- \(\Phi\) is magnetic flux (Tm
^{2}or Wb) - \(B \) is magnetic flux density (T)
- \(A\) is the area of the surface
- \(\theta\) is the angle between the magnetic field lines and the normal (perpendicular) to \(A\)

### Faraday's law

#### Definition

Faraday's law states that the magnitude of the induced EMF is proportional to the rate of change of magnetic flux.

\(|\varepsilon |\propto{\mathrm{d}\Phi\over \mathrm{d}t}\)

- \(|\varepsilon |\) is the magnitude of the induced EMF (V)
- \(\Phi\) is the magnetic flux (Tm
^{2}or Wb) - \({\mathrm{d}\Phi\over \mathrm{d}t}\) is the rate of change of magnetic flux (Tm
^{2}s^{-1}or Wb s^{-1})

The magnitude of the induced EMF is equal to the rate of change of magnetic flux linkage, where magnetic flux linkage is the product of the number of turns on the conducting coil and the magnetic flux.

\(|\varepsilon |={\mathrm{d}N\Phi\over \mathrm{d}t}\)

- \(|\varepsilon |\) is the magnitude of the induced EMF (V)
- \(N\Phi\) is the magnetic flux linkage (Tm
^{2}or Wb) - \({\mathrm{d}N\Phi\over \mathrm{d}t}\) is the rate of change of magnetic flux linkage (Tm
^{2}s^{-1}or Wb s^{-1})

Learners often find the differentiation symbols intimidating - but generally you won't need to consider instantaneous moments in a continually changng situation. Instead, think of the *rate of chage of flux linkage *as:

\({\text{final flux linkage }-\text{ initial flux linkage}}\over \text{time}\)

#### Moving wire

Instead of having a plane area within the field, the alternative is a length of wire that sweeps through it.

\(\varepsilon=BLv\sin\theta\)

- \(\varepsilon\) is the EMF induced (V)
- \(B \) is magnetic flux density (T)
- \(L\) is the length of wire (m)
- \(v\) is the velocity of the wire (ms
^{-1}) - \(\theta\) is the angle between the magnetic field lines and the normal (perpendicular) to the swept out area

The EMF induced is constant. A real-world example is a plane flying through the Earth's magnetic field!

The subject guide suggests that all cases of a wire moving through the field will be at *right angles*. But there's no harm in being prepared!

## Stretch for 7

#### Derivation

It is interesting, but not essential, to know how to derive Faraday's law. You should revise AHL electric potential beforehand.

To do so, consider a metal wire. At the point at which the electric force acting on electrons (to return them to their evenly distributed state) is equal to the magnetic force (causing them to move to one side):

\(F_B=F_E\)

\(Bev=Ee\Rightarrow Bev={V\over L} e\)

\(V=BLv\)

- \(V\) is the work done per unit charge against the electric force by the magnetic force (V)
- \(B\) is magnetic flux density (T)
- \(L\) is the length of the conducting wire (m)
- \(v\) is the velocity at which the wire moves through the field (ms
^{-1})

Since the work done per unit charge in bringing the charges to their induced position is equal to that which would be released when a complete circuit is formed:

\(\varepsilon = BLv\)

- \(\varepsilon\) is the induced EMF (V)

Or if the wire is replaced by a coil with multiple turns:

\(\varepsilon=BLvN\)

- \(N\) is the number of turns in the coil

### Applications

#### Generating electricity

Electromagnetic induction is the principle on which almost all methods for generating electricity are founded. All of the following contain a turbine that turns a coil relative to a magnetic field (or vice versa):

- Fossil fuel power stations
- Nuclear power stations
- Wind turbines
- Hydroelectric turbines
- Wave and tidal turbines
- Geothermal

Check out Energy sources if you need a recap!

#### Other uses

The following devices use electromagnetic principles:

- Transformers
- Electromagnetic braking - friction brakes (which dissipate kinetic energy as thermal energy) can be replaced by electromagnets (which convert the kinetic energy into electrical energy). A benefit is that these do not wear out and so do not require replacement.
- Geophones - seismic events cause a magnet to move relative to a coil. An inertial mass on a spring is connected rigidly either to the magnet or the coil to ensure that one remains stationary. The greater the magnitude of the earthquake, the greater the voltage induced.
- Metal detectors - when a current-carrying coil moves relative to a conductor beneath the ground, its magnetic field induces an EMF in the hidden conductor. The hidden conductor in turn becomes an electromagnet and exerts a force on the original coil.