# Interactive textbook: Magnetism

### It's not what you think

When you think about magnetism you naturally think about horseshoe magnets, North and South poles, compasses and fridges. But to understand the fundamental nature of magnetism we need to look at something different: moving charges. Let's start with this and come back to fridge magnet later.

### Magnetic force

Magnetism is related to moving charges. Two current-carrying conductors experience a force:

The **magnetic** or Lorentz force:

- is not the electric force as the wires have no overall charge
- has magnitude that depends on the size of the current
- has direction perpendicular to the direction of current (i.e. towards the other wire)
- is not a fundamental force but is instead a consequence of the electric force for moving charges.

To model this effect we use the idea of a magnetic field and, to understand where that comes from, we need to go back to magnets and compasses.

Which of the folowing are true?

Magnets have north and south poles but they can not be separated.

The magnetic force is in the opposite direction to electric force.

### Magnetic field

Magnetism was discovered long before its connection to electricity was made.

A **magnetic field** is a region of space where a small magnet (e.g. compass needle) would experience a turning force. The direction of the field is the direction that the North-seeking pole would point.

The reason that a compass needle points North is that it also has a magnetic field around it. The magnetic field of the compass lines up with the magnetic field of the Earth. By placing compasses around the Earth we can draw magnetic field lines to enable us to visualise the field.

If we were to take a smaller compass near the big one it would line up with the field of the big needle we could also plot the field around it.

Notice that the patterns are very similar - except that the magnetic North-seeking pole of the Earth's magnet must be in the southern hemisphere!

To determine that a nail is a magnet you could show that:

If one end attracts to the N and the other to the S it could be unmagnetised steel.

If it attracts another nail the other one could be the magnet

**Bar magnets**

The compass needle is a small bar magnet with poles at either end. **Magnetic field lines** that show the direction that a North-seeking small magnet would point if placed in the field.

We can use iron filings or plotting compasses to observe that the field lines go from N to S. However, they are actually continuous, forming loops through the magnet.

This is an important difference between magnetic and electric field lines:

- Electric field lines start on negative charges and end on positive ones
- Magnetic field lines don't have a start and end; they are continuous loops.

This means that if you take any area of space the number of magnetic field lines entering it will always equal that leaving it.** **However if there is a net positive charge in the area, the electric field lines entering will be less than those leaving. These are related to two of Maxwell's equations which are fundamental to electromagnetism but not in the Subject Guide.

If more magnetic field lines enter the end of a magnet than leave it:

field lines in = field lines out, it is not possible to have a N or S pole on its own unlike in electricity where you can have + and - particles.

**Uniform field**

A **uniform** magnetic field can be created between two flat magnets. The uniform section is indicated by equally spaced, parallel lines.

#### Straight wire

The magnetic field around a current-carrying straight wire can be plotted using a small compass.

The compass always points at a tangent to a circle centred on the conductor. The field pattern looks like this if viewed from above.

Where is the N pole in this magnetic field?

The field lines are continuous there in no N and S.

Is it possible to create an electric field with the same pattern of field lines?

Electric field lines start on + charges end on - charges.

Notice how the field lines get further apart to indicate that the field is getting weaker. The dot in the middle represents current coming towards you, current away is a **x**, like an archer's arrow**.**

The diagram represents a magnetic field.

The field is:

The dots are not equally spaced.

A dot represents an arrow pointing towards you.

The direction of the field is anticlockwise. To help you to remember this you can use the right-hand grip rule.

The thumb is in the direction of the current and the fingers point in the direction of field.

**Coils**

When a wire is made into a coil, the sum of magnetic fields due to each section of current around the edge creates a resultant field pattern as shown.

The grip rule can also be used to find the direction here.

If the hand was placed at the top of the coil the thumb would point

The thumb would still point in the direction of the current.

**Solenoid**

A solenoid is a coil wound so that each turn lies next to the others rather than on top of each other as in a coil.

This can be drawn more simply in two dimensions.

Note the direction of the current; applying the grip rule at the end gives the direction of the field through the solenoid. Also note that the field at the centre is uniform (in this case to the left).

The diagram represents the end of a solenoind.

If you were to draw the magnetic field lines for this situation they would be:

the field goes into this end

If a steel bar was placed in this solenoind this end would become

field line go from N - S

The field inside a solenoind is

At the centre the field lines are parallel and equally spaced

### Current-carrying conductors

When two magnets are placed close to each other they either attract or repel depending on which ends are near each other.

If the field lines are in the same direction they attract. When a current-carrying conductor is placed perpendicular to a uniform magnetic field, the field lines are in the same direction on top and the opposite direction below:

The fields in the diagram above add vectorially. The field will be

The field lines add above and cancel below.

This causes the conductor to experience a force downwards perpendicular to both current and field. To help us to quickly work out which way the force is we can use Fleming's left hand rule.

When applying this rule you must keep your fingers stiff as you rotate your hand (or examination paper) so that your first finger points in the direction of the field cutting the conductor and your second finger point in the direction of conventional current. Your thumb will then give the direction of the force.

### Definition of the ampere

Although it would seem sensible to define the amp in terms of charge per unit time, it is defined in terms of the force between two parallel conductors.

If we view these conductors from above we can see that each conductor is in the field of the other. Here X is shown to be in the field of Y.

The field through X is in the direction of the tangent to the circle so points vertically down. Using Fleming's left hand rule shows the direction of the force is to the right.

If you repeat the same process with Y you find it experiences a force to the left. This must be the case due to Newton's 3rd law. If Y pulls X to the right, X must pull Y to the left with a force of equal magnitude. The conductors attract each other.

\(1\text{ A}\) is the current that would cause a force of \(2 \times 10^{-7}\text{ Nm}^{-1}\) between two long parallel conductors placed \(1\text{ m}\) apart in a vacuum.

If the hand was placed on the blue wire the thumb would point:

First in line with field opposite to the direction of the field experienced by the red.

Second in line with the current, same as the red

### Magnetic flux density

We know that the proximity of field lines tells us how strong the field is but so far we haven't defined a quantity equivalent to field strength in electric and gravitational fields. The field strength in an electric field was defined in terms of the force per unit charge on a positive test charge. To use a similar approach we would have to measure the force on a North pole... but North poles don't exist on their own.

We define the strength of the magnetic field in terms of the force on a current-carrying conductor. So the **flux density** (magnetic field strength) is the force per Amp per meter experienced by a current-carrying conductor placed perpendicular to the field.

\(B={F\over IL}\)

- \(B\) is magnetic flux density in Tesla (\(\text{T}\))
- \(F\) is magnetic force in \(\text{N}\)
- \(I\) is current in \(A\)
- \(L\) is length in \(\text{m}\)

If the flux density is \(1\text{ T}\) then the force on a \(1\text{ m}\) long wire carrying \(1\text{ A}\) perpendicular to the field is \(1\text{ N}\).

#### Force on a current-carrying conductor

Now, if we know the flux density we can calculate the force on a wire using the rearrangement:

\(F=BIL\)

A 20 cm long cylindrical conductor is placed at an angle of 30° to a uniform magnetic field of flux density 10 T.

What is the component of the field perpendicular to the wire

B sin 30°

If the current in the wire is 3 A calculate the force experienced by the wire

F = BIL = 5 x 3 x .2

#### Force on a moving charge

Consider a section of wire in a perpendicular magnetic field:

The magnetic force on the conductor is given by the equation:

\(F=BIL\)

The current is due to the drift of electrons:

\(I=nAve\)

\(\Rightarrow F=BnAveL\)

\(n\) is the number of free electrons per unit volume, so the number of free electrons in this cylinder is \(nAL\).

The force on each electron is therefore:

\(F_e={BnAveL\over nAL}=Bev\)

More generally, if a charge \(q\) travels at velocity \(v\) perpendicular to a magnetic field of flux density \(B\), it will experience a force \(F\) perpendicular to its direction of motion.

\(F=Bqv\)

Note the direction of the force:

- If the charge is positive, it travels in the direction of conventional current so your second finger points up.
- If the charge is negative your second finger must point in the opposite direction to velocity.

This force is always perpendicular to the direction of motion so the charged particle will move in a circle.

The radius, \(r\), of this circle can be found by equating the equation for centripetal force with the equation for magnetic force:

\({mv^2\over r}=Bqv\)

\(r={mv^2\over Bqv}={mv\over Bq}\)

If the charge is projected into the field at an angle, a helical path results.

If the field got stronger towards the top of the diagram the circles of the helix would get;

The radius is inversley proportional to B.

The speed would not change so the circles would be completed in less time and the charges would not progress so far upwards.