This notes sheet looks at magnetic fields & electromagnetism, including induction, the magnetomotive force, permeability, reluctance and electromagnets.

## Magnetic Fields

**A magnetic field is a vector field in which a magnetic material experiences a force. **There are two

causes of magnetic fields:

**Permanent Magnets**

**Moving Charge**- This could be around a straight,
**current-carrying wire**:

- This could be around a straight,

- A
**flat coil**:

- or a
**solenoid:**

### Flux

**Flux Density, B**, is a measure of field strength, the density of field lines.

The units of flux are Tesla, T (N/Am).

**Magnetic flux, φ**, is the flow of the field, represented as field lines.

- For fixed magnets, flux always goes from north to south
- For a current-carrying wire, the right-hand grip rule shows the direction of flux
- Like poles repulse, opposite poles attract
- Magnetic flux lines never cross

The units of flux are Weber, Wb.

Fl**ux Linkage, Nφ **, applies when a wire is coiled, for example a solenoid. In this instance, the flux

linkage is the flux multiplied by the number of turns:

The units of flux linkage are Weber turns.

## Induction

When a current carrying wire is in a magnetic field, it experiences a force. This is known as the**Lorentz Force**, and is given as:

The direction of the force is found using **Fleming’s left-hand rule**:

Alternatively, if a charged particle moved through the magnetic field with velocity , it expereinces a force:

### Faraday’s Law

According to Faraday’s Law, an emf is induced in the circuit whenever it experiences a change in

magnetic flux linkage:

In a closed circuit, this emf causes a current to flow.

Generally, the change of flux occurs because the wire/circuit is moving. In this case, the direction

of the induced current is found using the **right-hand rule**:

### Lenz’s Law

According to Faraday’s Law, when a wire moves through a magnetic field, a current is induced.

This current opposes the current that caused the initial Lorentz force that moved the wire.

This means all generated power has resistance and is known as **Lenz’s Law**.

## Electromagnetic Quantities

**Magnetism is very similar to electricity** – for every electrical property (emf, current, resistance etc.),

there is an equivalent electromagnetic quantity that performs in exactly the same way. Electric

equations and relations, like Ohm’s law, have their direct equivalents too.

### Magnetomotive Force, mmf

Just like electrical current is the flow of electromotive force (emf) around a circuit, magnetic flux is

the flow of magnetomotive force around a circuit.

A solenoid, or any coil of current-carrying wire is an mmf source – equivalent to an emf source in

an electrical circuit. The magnitude of the mmf delivered is given as:

- is the number of turns
- is the current in the wire

### Reluctance, S

This is the magnetic equivalent to electrical resistance. Therefore, magnetic Ohm’s Law is:

When connected in series, reluctances add up like resistors:

Similarly in parallel:

A key difference in magnetic circuits, however, is that reluctance is not given in a ‘reluctor’ in the

way that a resistance comes from a resistor.

Instead, reluctance rises from cores made of magnetic materials. The reluctance depends on the

material, cross-section, and length:

- is the length of the core
- is the cross-sectional area

### Permeability

Magnetic Permeability is a material property. It is defined relative to the permeability of free space:

- is the relative permeability of the material
- is the permeability of free space

**Ferromagnetic **materials have a high relative permeability, making them excellent magnetic

conductors. They are easy to magnetise and magnetise and are often called **soft magnets**. They

are called ferromagnetic, as they generally have high iron content.

Other magnetic materials are **hard**, however: it is significantly harder to magnetise and

demagnetise these.

### Comparing Electrical and Magnetic Quantities

Electrical Quantity | Units | Magnetic Quantity | Units |
---|---|---|---|

emf | mmf | ||

Current, | Flux, | Wb | |

Resistance, | Reluctance, | ||

Resistivity, | 1/Permeability, |

Know these equivalents, and magnetic circuits will be a breeze.

## Energy in Magnetic Circuits

Energy in a magnetic circuit is given as:

This can be written in many forms:

### Air Gaps

Many magnetic circuits have air gaps. The relative permeability of these is 1, so the reluctance in an air gap is just:

Where is the thickness of the gap.

This reluctance is significantly larger than that of the core, and the two reluctances are in series:

where

Therefore, the vast majority of the energy stored in the circuit is in the air gap:

Since work done is force times distance moved:

This is how **electromagnets **work: an electric winding on a core produces a vast force in a small

air gap between the core and a magnetic object (the armature). If this force is greater than the

frictional and gravitational forces holding the armature stationary, it will move towards the

electromagnet. There are two currents:

**Pull-in current**

This is the current required to move the armature over the air gap**Holding current**

This is the current required to hold the armature once it has made contact with the core

(there is no air gap)

The holding current is significantly lower than the pull-in current.

- A magnetic field is a vector field in which a magnetic object experiences a force
- They are either caused by permanent magnets, or by moving charges

**Flux density**, , is a measure of field strength. The units are Tesla, T.**Flux**, , is the flow of the field- The
**Lorentz Force**is experienced by- a current carrying wire in a magnetic field:
- a charged particle moving in a magnetic field:

- Electric quantities all have their magnetic counterpart – see above
- The energy in a magnetic circuit is given as
- The energy stored in an air gap is far higher than that stored in the core, as the reluctance is so much higher