Magnetic Fields and the Force on a Current-Carrying Conductor

A wire carrying a current in a magnetic field feels a force. That single fact drives every electric motor, loudspeaker and moving-coil meter, and it is how physicists define the strength of a magnetic field. This note introduces magnetic fields and field lines, the equation $F = BIL\sin\theta$, Fleming's left-hand rule and the definition of magnetic flux density. All of it is Paper 4 material, and the current-balance experiment for measuring $B$ is a favourite Paper 5 planning context.

What a magnetic field is

You have already met two fields of force: gravitational fields, which act on masses, and electric fields, which act on charges. A magnetic field is a third kind.

The two sources are really one. Inside a permanent magnet, the field comes from the motion and spin of electrons in the atoms; a current in a wire is also charge in motion. Every magnetic field ultimately comes from moving charge.

A stationary charge in a magnetic field feels no magnetic force. This is a key difference from electric fields, which act on charges whether they move or not.

Representing a field with field lines

Magnetic fields are drawn with field lines (also called lines of magnetic flux). The rules are:

• The direction of a field line at a point is the direction a small compass needle would point: the direction of the force on a north pole placed there.
• Outside a magnet, field lines run from the north pole to the south pole. Inside the magnet they continue from south to north, so each line is a closed loop.
• Field lines never cross: the field has only one direction at each point.
• The closer the lines, the stronger the field. Equally spaced parallel lines mean a uniform field.

The field of a bar magnet spreads out from the north pole, curves round and converges on the south pole; it is strongest near the poles where the lines crowd together. Between the flat, parallel faces of a north pole and a south pole placed close together (as in the jaws of a U-shaped Magnadur magnet), the field is very nearly uniform: straight, parallel, equally spaced lines from N to S, with some bulging at the edges.

Into and out of the page

Most exam diagrams are two-dimensional, so fields perpendicular to the paper need a symbol:

Symbol	Meaning	Memory aid
$\times$ (cross)	field (or current) directed into the page	the tail feathers of an arrow moving away from you
$\bullet$ (dot)	field (or current) directed out of the page	the point of an arrow coming towards you

A region filled with evenly spaced crosses is a uniform field into the page.

The motor effect

Place a straight wire between the poles of a magnet so that it crosses the field lines, and pass a current through it. The wire is pushed sideways. This is the motor effect.

Why does it happen? The current produces its own magnetic field: circles around the wire. On one side of the wire this circular field is in the same direction as the magnet's field, so the fields add and the resultant field is strong. On the other side the two fields are opposite and partly cancel, so the resultant field is weak. The wire is pushed from the strong-field side towards the weak-field side. (This combined pattern is sometimes called a catapult field: the field lines behave as if stretched and pushing the wire out.)

Three facts follow from experiment:

1. The force is perpendicular to both the current and the field.
2. Reversing either the current or the field reverses the force. Reversing both leaves the force unchanged.
3. If the wire is parallel to the field, there is no force at all.

Fleming's left-hand rule

Hold the thumb, first finger and second finger of your left hand mutually at right angles.

• First finger: Field (from N to S).
• Second finger: Current (conventional current, from $+$ to $-$).
• Thumb: thrust, the force (motion).

Use the rule mechanically. Point the first finger along $B$, rotate the hand until the second finger points along $I$, and read the force from the thumb.

The size of the force: F = BIL sin θ

Experiments with a wire between magnet poles show that the force is proportional to:

• the current $I$,
• the length $L$ of wire inside the field,
• the strength of the field,
• $\sin\theta$, where $\theta$ is the angle between the wire (current) and the field.

The strength of the field is the constant of proportionality, $B$, called the magnetic flux density.

Only the component of the field perpendicular to the wire, $B\sin\theta$, pushes on the current. The component parallel to the wire does nothing.

Magnetic flux density and the tesla

Rearranging $F = BIL$ for a wire at right angles to the field gives $B = F/IL$. This is the definition the syllabus uses.

A flux density of one tesla is large. Some typical values:

Source	Approximate $B$
Earth's field at the surface	$5 \times 10^{-5}\ \text{T}$
Small Magnadur magnets in a school laboratory	$0.05$ to $0.2\ \text{T}$
Strong neodymium magnet near its surface	$0.5$ to $1\ \text{T}$
Hospital MRI scanner	$1.5$ to $3\ \text{T}$

Forces on a rectangular coil

A rectangular coil in a uniform field is the heart of an electric motor. With the plane of the coil parallel to the field:

• The two sides perpendicular to the field each feel a force $F = NBIL$, in opposite directions (the current flows in opposite directions along them). These two forces form a couple and turn the coil.
• The two sides parallel to the field feel no force.

As the coil turns, the forces on the long sides stay the same size (those sides remain perpendicular to $B$), but their lines of action move closer together, so the turning effect falls to zero when the plane of the coil is perpendicular to the field. A motor uses a split-ring commutator to reverse the current at that point so the coil keeps turning the same way.

Measuring magnetic flux density with a current balance

The direction of the change tells you the direction of the force. If the balance reading increases, the magnet is being pushed down, so the wire is being pushed up.

Worked examples

Practice questions

Answers

1. Magnetic flux density is the force acting per unit current per unit length on a wire placed at right angles to the magnetic field. $1\ \text{T} = 1\ \text{N A}^{-1}\ \text{m}^{-1} = 1\ \text{kg m s}^{-2}\ \text{A}^{-1}\ \text{m}^{-1} = 1\ \text{kg s}^{-2}\ \text{A}^{-1}$.
2. $F = BIL = 0.12 \times 5.0 \times 0.25 = 0.15\ \text{N}$.
3. $B = F/IL = 0.024/(4.0 \times 0.060) = 0.10\ \text{T}$.
4. $F = BIL\sin\theta = 0.050 \times 6.0 \times 0.30 \times \sin 40^\circ = 0.058\ \text{N}$. The maximum is at $90^\circ$: $F = 0.050 \times 6.0 \times 0.30 = 0.090\ \text{N}$.
5. $\Delta m\, g = BIL$, so $B = g \times \text{gradient}/L$. Gradient $= 0.48 \times 10^{-3}\ \text{kg A}^{-1}$: $B = 9.81 \times 0.48 \times 10^{-3}/0.050 = 0.094\ \text{T}$.
6. The rod is horizontal and perpendicular to the field (vertical), so $F = BIL = 0.25 \times 5.0 \times 0.30 = 0.375\ \text{N}$, and $a = F/m = 0.375/0.040 = 9.4\ \text{m s}^{-2}$. The force is perpendicular to the field, which is vertical, so the force is horizontal: it pushes the rod along the rails, never up.
7. $BIL = mg$: $I = mg/BL = (6.0 \times 10^{-3} \times 9.81)/(0.080 \times 0.15) = 4.9\ \text{A}$.
8. The field lies in the north–vertical plane, and the wire runs east–west, so the wire is at $90^\circ$ to the field. $F = BIL = 5.0 \times 10^{-5} \times 6.0 \times 0.50 = 1.5 \times 10^{-4}\ \text{N}$. The force is perpendicular to the wire, so it lies in the north–vertical plane, and perpendicular to the field. Applying the left-hand rule to each component: the downward component of $B$ ($B\sin 65^\circ$) with eastward current gives a force towards the north; the northward component ($B\cos 65^\circ$) gives a force upwards. The resultant points north and upwards, at $25^\circ$ above the horizontal (at right angles to the field, which is $65^\circ$ below the horizontal).