Magnetic flux density and magnetic forcesEdexcel International A Level Physics: Revision notes

A magnetic field is represented by field lines. The magnetic flux density $B$ measures how strong the field is. It is defined using the force on a current-carrying wire: when a wire of length $l$ carries a current $I$ perpendicular to the field, the force is $F = BIl$, so $B = \frac{F}{Il}$. The unit is the tesla (T), where 1 T = 1 N A⁻¹ m⁻¹. One tesla gives a force of 1 N on 1 m of wire carrying 1 A perpendicular to the field.

The magnetic flux $\Phi$ through an area $A$ perpendicular to the field is $\Phi = BA$. The unit is the weber (Wb), where 1 Wb = 1 T m². If the normal to the area makes an angle $\theta$ with the field, only the perpendicular component counts, so $\Phi = BA\cos\theta$.

For a coil of $N$ turns, each turn is linked by the same flux, so the flux linkage is $N\Phi$ (unit: Wb turns, or simply Wb).

A charge $q$ moving at speed $v$ at angle $\theta$ to a field of flux density $B$ experiences a force

$F = Bqv\sin\theta$

The force is largest ($F = Bqv$) when the velocity is perpendicular to the field and zero when the charge moves parallel to the field.

The force is always perpendicular to the velocity, so it does no work and the speed and kinetic energy stay the same. It changes only the direction of motion. With the velocity perpendicular to a uniform field the force provides the centripetal force, so the path is a circle.

Fleming's left-hand rule gives the direction of the force. Hold the thumb, first finger and second finger of the left hand at right angles:

• first finger points along the field (north to south pole)
• second finger points along the conventional current
• thumb points along the force (motion)

For a moving positive charge, the second finger points along its velocity. For a negative charge such as an electron the conventional current is opposite to its velocity, so point the second finger opposite to the velocity (or find the force for a positive charge and reverse it).

A conductor of length $l$ carrying current $I$ at angle $\theta$ to a uniform field experiences

$F = BIl\sin\theta$

The force is maximum when the wire is perpendicular to the field and zero when it is parallel. It arises because the forces on the moving charge carriers add up along the wire. Use Fleming's left-hand rule for its direction.

In a current balance the equal and opposite force on the magnet is measured as a change in reading, so $B = \frac{F}{Il}$ can be found.

Question. An electron moves at $4.0 \times 10^6$ m s⁻¹ at 90° to a field of 0.050 T. Find the force.

$F = Bqv = 0.050 \times 1.60 \times 10^{-19} \times 4.0 \times 10^{6} = 3.2 \times 10^{-14}$ N

Question. A 0.20 m wire carries 5.0 A at 30° to a field of 0.40 T.

$F = BIl\sin\theta = 0.40 \times 5.0 \times 0.20 \times \sin 30^\circ = 0.20$ N

Always state the angle you are using and check it is between the velocity (or wire) and the field.