IMAT Physics · Magnetism

Magnets, Magnetic Fields and the Force on Currents and Charges

Magnets and currents make magnetic fields; a field pushes sideways on a current or a moving charge, which can bend a charged particle into a circle.

All Magnetism notes

Why this matters

The 2014 paper asked what kind of bar a magnet could repel. The 2024 ministry paper asked about the circular path of an electron that enters a magnetic field at right angles, asking which statement about it is false.

Concept 1 of 5: Magnets, poles, field lines and the Earth's field

Every magnet has a north pole and a south pole, and you can never get one without the other: cut a magnet in half and you get two smaller magnets. Like poles push apart and unlike poles pull together. Iron, nickel and cobalt are pulled by any magnet because the magnet turns them into temporary magnets, with the nearer end opposite to the pole that is close.

Definition

Poles: like poles repel, unlike poles attract. Poles always come in N and S pairs.
Field lines outside a magnet run from N to S. They never cross; where they are closer together the field is stronger. A compass needle lines up along them.
Ferromagnetic materials (iron, nickel, cobalt and alloys such as steel) are attracted to either pole. Soft iron magnetises and demagnetises easily (electromagnet cores); steel keeps its magnetism (permanent magnets).
Repulsion is the only proof that an object is a magnet: an unmagnetised piece of iron is attracted by both poles, but never repelled.
The Earth behaves like a giant bar magnet with field about $5 \times 10^{-5}\ \text{T}$ . A compass's N pole points to geographic north, so the Earth's magnetic pole there is a south pole.
Magnetic field strength $B$ is measured in tesla (T).

Object near a bar magnet	Force it feels	Reason
Unmagnetised iron, nickel or cobalt	Attraction to either pole, never repulsion	The magnet induces an opposite pole in the near end
Another magnet (e.g. magnetised steel)	Attraction or repulsion, depending on which poles face	Like poles repel, unlike poles attract
Copper, aluminium, wood, plastic	Practically none	Not ferromagnetic
An electrically charged object at rest	No magnetic force	A magnetic field acts only on moving charges

Practice this conceptself-check · 3 quick reps

Attraction does not prove an object is a magnet

Any piece of iron, nickel or cobalt is attracted by a magnet, magnetised or not. Only repulsion shows the object is a magnet. A charged but non-magnetic object at rest feels no magnetic force at all.

Concept 2 of 5: The magnetic field of a straight wire and of a solenoid

A current makes a magnetic field that wraps round the wire in circles. The field is stronger near the wire and with more current. Wind the wire into a long coil (a solenoid) and the circles from all the turns add up inside it, giving a strong, nearly uniform field like that of a bar magnet.

Definition

Straight wire: the field lines are circles centred on the wire. The right-hand grip rule: thumb along the current, curled fingers give the field direction. $B = \mu_0 I/(2\pi r)$ , so $B \propto I$ and $B \propto 1/r$ .
Solenoid: inside, the field is uniform and along the axis, $B = \mu_0 n I$ , where $n = N/L$ is the number of turns per metre. Outside it looks like the field of a bar magnet. Grip the coil with the fingers along the current: the thumb points to the N end.
$\mu_0 = 4\pi \times 10^{-7}\ \text{T m/A}$ . An iron core makes the field much stronger (an electromagnet).
Reversing the current reverses the field.

Field of a long straight wire and of a solenoid

B_{\text{wire}} = \frac{\mu_0 I}{2\pi r} \qquad B_{\text{solenoid}} = \mu_0 \frac{N}{L} I

$B$ magnetic field strength, in T
$I$ current, in A
$r$ distance from the wire, in m
$N/L$ turns per metre of the solenoid
$\mu_0$ $4\pi \times 10^{-7}\ \text{T m/A}$

Worked example

A long straight wire carries 10 A. Find the magnetic field 5.0 cm from it and compare it with the Earth's field of about

5 \times 10^{-5}\ \text{T}

Practice this conceptself-check · 3 quick reps

A wire's field falls as 1/r, not 1/r²

The field of a long straight wire is proportional to

1/r

: twice as far, half the field. The inverse square law belongs to point charges and point masses. Using it for a wire gives the wrong option every time.

Concept 3 of 5: The force on a current-carrying wire in a magnetic field

A current in a magnetic field feels a push at right angles to both the current and the field. This is the motor effect: it is what turns an electric motor. The push is largest when the wire crosses the field at right angles and disappears when the wire runs along the field.

Definition

The force on a straight wire of length $L$ carrying current $I$ at angle $\theta$ to a field $B$ is $F = BIL\sin\theta$ .
Maximum when the wire is perpendicular to the field ( $\sin 90^\circ = 1$ ); zero when it is parallel ( $\sin 0^\circ = 0$ ).
Direction: Fleming's left-hand rule. First finger along the Field, seCond finger along the Current, thuMb gives the Motion (force). The force is perpendicular to both $B$ and $I$ .
Two parallel wires with currents in the same direction attract; opposite directions repel.
This also defines the tesla: $1\ \text{T} = 1\ \text{N}/(\text{A m})$ .

Force on a wire

F = B I L \sin\theta

$F$ force, in N
$B$ magnetic field, in T
$I$ current, in A
$L$ length of wire in the field, in m
$\theta$ angle between the wire and the field

Worked example

A 0.25 m length of wire carries 4.0 A in a uniform field of 0.30 T. Find the force on it when it is at right angles to the field, and when it makes 30° with the field.

Practice this conceptself-check · 3 quick reps

No force when the current runs along the field

The force depends on

\sin\theta

. A wire lying parallel to the field feels no force at all, however large the current or the field. The force is never along the field lines; it is always at right angles to them.

Concept 4 of 5: The force on a moving charge: F = qvB sinθ

A current is just moving charges, so each moving charge in a field feels its own sideways push. Because the push is always at right angles to the motion, it can turn the particle but can never speed it up or slow it down. A charge at rest, or one moving along the field, feels nothing.

Definition

A charge $q$ moving at speed $v$ at angle $\theta$ to a field $B$ feels $F = qvB\sin\theta$ .
The force is perpendicular to the velocity (and to the field). So it does no work: the speed and kinetic energy stay constant; only the direction changes.
A charge at rest, or moving parallel to the field, feels no magnetic force. A neutral particle (neutron) feels none either.
Direction: use Fleming's left-hand rule with the second finger along the direction of conventional current. For an electron, the current is opposite to its motion, so the force is reversed.

Force on a moving charge

F = q v B \sin\theta

$q$ charge, in C
$v$ speed, in m/s
$B$ magnetic field, in T
$\theta$ angle between the velocity and the field

Worked example

A proton moves at

2.0 \times 10^6\ \text{m/s}

at right angles to a magnetic field of 0.50 T. Find the force on it. What would the force be on an electron moving the same way at the same speed?

Practice this conceptself-check · 3 quick reps

A magnetic field cannot change a particle's speed

The magnetic force is always at right angles to the velocity, so it does no work. The kinetic energy and speed stay the same; only the direction turns. To speed a charge up you need an electric field.

Concept 5 of 5: Circular motion of a charged particle in a magnetic field

A force that always pushes at right angles to the motion, with a constant size, is exactly what keeps something going round a circle. So a charge fired at right angles into a uniform field moves in a circle at constant speed. A faster particle swings round a wider circle, but it also travels faster, so the time for one lap stays the same.

Definition

For a charge entering a uniform field at right angles:

The magnetic force provides the centripetal force: $qvB = mv^2/r$ , so $r = mv/(qB)$ .
The motion is uniform circular motion: constant speed, but the velocity changes all the time because its direction changes.
The period $T = 2\pi r/v = 2\pi m/(qB)$ and the frequency $f = qB/(2\pi m)$ do not depend on the speed.
Heavier or faster particles, weaker fields and smaller charges give wider circles.

Radius and period of the circle

r = \frac{m v}{q B} \qquad T = \frac{2\pi m}{q B}

$r$ radius of the circle, in m
$m$ mass of the particle, in kg
$v$ speed, in m/s
$T$ time for one revolution, in s

Worked example

A proton (

m = 1.67 \times 10^{-27}\ \text{kg}

q = 1.6 \times 10^{-19}\ \text{C}

) enters a 0.20 T field at right angles with a speed of

3.0 \times 10^6\ \text{m/s}

. Find the radius of its path and the time for one revolution.

Practice this conceptself-check · 3 quick reps

Constant speed is not constant velocity

In a magnetic field the speed stays fixed but the direction turns all the time, so the velocity (a vector) is always changing. Any statement that the particle keeps the same velocity after entering the field at right angles is false.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

The magnetic field of a straight wire and of a solenoid
Field of a long straight wire and of a solenoid
$B_{\text{wire}} = \frac{\mu_0 I}{2\pi r} \qquad B_{\text{solenoid}} = \mu_0 \frac{N}{L} I$
The force on a current-carrying wire in a magnetic field
Force on a wire
$F = B I L \sin\theta$
The force on a moving charge: F = qvB sinθ
Force on a moving charge
$F = q v B \sin\theta$
Circular motion of a charged particle in a magnetic field
Radius and period of the circle
$r = \frac{m v}{q B} \qquad T = \frac{2\pi m}{q B}$

Reference tables (1)

Magnets, poles, field lines and the Earth's field4 rows

Object near a bar magnet	Force it feels	Reason
Unmagnetised iron, nickel or cobalt	Attraction to either pole, never repulsion	The magnet induces an opposite pole in the near end
Another magnet (e.g. magnetised steel)	Attraction or repulsion, depending on which poles face	Like poles repel, unlike poles attract
Copper, aluminium, wood, plastic	Practically none	Not ferromagnetic
An electrically charged object at rest	No magnetic force	A magnetic field acts only on moving charges

Watch out for (5)

Attraction does not prove an object is a magnet
→ Magnets, poles, field lines and the Earth's field
A wire's field falls as 1/r, not 1/r²
→ The magnetic field of a straight wire and of a solenoid
No force when the current runs along the field
→ The force on a current-carrying wire in a magnetic field
A magnetic field cannot change a particle's speed
→ The force on a moving charge: F = qvB sinθ
Constant speed is not constant velocity
→ Circular motion of a charged particle in a magnetic field

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