# Magnetic Fields and Magnetism

This is a topic from Higher Physics 1B

## Contents |

## Introduction

Magnetism is a difficult concept to define explicitly, but one definition is as follows. Magnetism is a property of materials that can be influenced by the presence of a magnetic field, which is a type of 'field of influence' caused by moving charges and magnetic materials.

### Magnetic Poles

Every magnet, regardless of its shape or size, has two poles, North and South which exert forces on one another in a similar manner to electric charges - similar poles repel, opposite poles attract. Magnetic poles always come in pairs (unlike charge, a single pole has never been isolated). The force between two poles is proportional to the inverse square of the distance between them.

## Introducing Magnetic Fields and Forces

### Magnetic Fields

Magnetic fields exist around permanent magnets and moving charges, and have the following properties:

- By definition current is moving charge, and so current induces a magnetic field; in this context magnetic field is often referred to as
*magnetic induction*. - Magnetic field is a vector quantity
- It is represented by the symbol
**B** - The unit is the Tesla in S.I units, or alternatively the Gauss (1 T = 10
^{4}G) - Magnetic field lines follow the same conventions as electric field lines - the direction of the lines reflects the direction a north pole would point, and the more dense the lines are the stronger the field

- By definition current is moving charge, and so current induces a magnetic field; in this context magnetic field is often referred to as

### Magnetic Forces

Magnetic force has the following properties:

- Any moving charge in an external magnetic field undergoes a force, because it induces its own magnetic field which interacts with the external one
- The magnitude is proportional to the charge, its speed and the magnitude of the magnetic field perpendicular to the charge
- The direction is dependant on the velocity of the charge and the direction of the magnetic field
- The direction of the magnetic force is perpendicular to the plane spanned by the vectors
**v**and**B** - It is given by the following equation:

**F**_{B}= qv x B**= qvB Sinθ**

Where F_{B} is the magnetic force in Newtons

- q is the charge in Coulombs
- v is the velocity of the moving charge in ms
^{-1} - B is the magnetic field in Teslas (which are equivalent to NsC
^{-1}m^{-1}by rearranging the equation)

### The Right-Hand Rule

The Right-Hand Rule is a tool for working out the direction of the magnetic force:

- The fingers of your right hand point in the direction of the velocity (for positive charges, the other way for negative charges)
- Curl fingers in the direction of B
- The thumb is pointing in the direction of the force:

### Differences between Electric and Magnetic Fields (Part 1)

Despite the many similarities of the two phenomenon, they have the following differences:

- Direction of force
- Electric force acts in the direction of the field
- Magnetic force acts perpendicular to the field

- Motion
- Electric force acts on a charged particles regardless of whether it is stationary or moving
- Magnetic force acts on a charged particle only when it is moving

- Work
- Electric force does work in displacing a charged particle
- Magnetic force from a constant magnetic field does no work in displacing a particle

- Direction of force

There are further similarities and differences with regard to concepts not covered here. See Part 2.

### Work in Magnetic Fields

- Because the magnetic force is in a direction perpendicular to the displacement of a charge it does no work on the charge (F•ds = 0)
- There is no work done, so the kinetic energy of the particle cannot by changed by the magnetic field alone
- Therefore the magnetic field can only change the direction of the particle - its speed is constant but its velocity changes

## Magnetic Fields and Current-Carrying Wires

- Magnetic force is exerted on each moving charge in the wire such that the total force equals the product of the force on one charge and the number of charges:

- F
_{B}= (qv x B)nAL

- F

Where n is the number of charges per unit volume (n = η / V where η is the total # of charges and V is volume)

- A is the cross-sectional area of the wire
- L is the length of the wire

- Analysing this:

- nqAv has units (ηm
^{-3})(C)(m^{2})(ms^{-1}) which simplifies to Cs^{-1}the unit of current

- nqAv has units (ηm

- This means force can be rewritten as:

- F = IL x B

Where I (nqAv) is current in Amperes (A)

- L is a vector with the direction of current flow and the magnitude of the length L of the segment

- For a wire of arbitrary shape we break L down into small segments ds such that the force is equal to the sum of the forces on these small segments:

- The above integral is equal to the vector sum of the small elements ds which is in effect the net displacement
- If a wire is curved, the value of the integral is the straight-line distance from the starting point to the finishing point
- If the wire is a closed loop, the vector sum of the segments ds is zero, as is the straight-line distance from start to finish - ie the force is zero

## Magnetic Fields and Charged Particles

- Magnetic force on a moving charged particle doesn't cause a change in speed, only in direction
- The direction of the force is always towards the centre of its subsequent circular direction, and thus the force is centripedal:

- F = qvB = mv
^{2}/ r

- F = qvB = mv

- Solving gives:

**r = mv / qB**such that the radius of the path is proportional to momentum and inversely preportional to magnetic field strength and charge

- The angular speed (or
*cyclotron frequency*)is given by:

- ω = v / r
- = qB / m

- Thus we can solve for the period of the motion:

- T = 2π / ω
- Therefore
**T = 2πm / qB**

### Charged Particles in Magnetic and Electric Fields

The total force equals the sum of the electric and magnetic component forces:

**F**_{Total}= F_{E}+ F_{B}- = qE + qv x B

### The Van Allen Belt

*You may be expected to understand what is meant by the term Van Allen Belt, which refers to a band of "cosmic ray" charged particles from the Sun which are trapped by the Earth's magnetic field and spiral from pole to pole, causing Auroras*

## Magnetic Flux

In general, ‘flux’ refers to the rate of flow per area, and so magnetic flux refers to the rate of flow of magnetic field lines through a given area (analogous to electric flux). Thus magnetic flux is the product of the magnitude of the magnetic field and the surface area perpendicular to it, as given by the equation:

Φ_{B} = BA

Where Φ represents magnetic flux in webers (Wb) or Tesla-metres-squared (Tm^{2})

- B represents the magnetic field (T)
- A represents perpendicular area (m
^{2})