Magnetic fields exert a force on a charge when the charge is moving. If the charge is stationary, no force is exerted. This force is given by:

${\overrightarrow {F}}=q({\overrightarrow {v}}\times {\overrightarrow {B}})$ ,

where q is the charge on the point charge, v is its velocity and B is the magnetic field strength. This involves a vector cross product, which you don't need to know about for A-level. However, you do need to know a simplified version of this. The magnitude of this force F is given by:

$F=Bqv\sin {\theta }$ ,

where θ is the angle between the direction of motion of the point charge and the direction of the magnetic field. If the velocity and the magnetic field are in the same direction, the θ = 0, so sin θ = 0 and F = 0. If the velocity and the magnetic field are perpendicular to each other, θ = ½ π, so sin θ = 1. This means that, in the special case where velocity is perpendicular to the magnetic field:

$F=Bqv$ If q is negative (for example, for an electron), the force is in the opposite direction.

## Current

A current is just a flow of moving electrons, and so a magnetic field will exert a force on a wire with a current flowing through it. The case you need to know about is when the magnetic field is perpendicular to the wire. In this case, the magnitude of the force on the wire is given by:

$F=BIL$ ,

where I is current, and L is the length of the wire.

## Direction

The direction of the force on either a point charge or on a wire can be worked out using Fleming's left-hand rule, as shown in the diagram on the right. The direction of the thumb is that of the force (or thrust), the direction of the first finger is that of the magnetic field, and the direction of the second finger is that of the current (or the motion of the point charge).

On a 2D diagram, the direction of a magnetic field is represented by one of two symbols, which resemble the point and fletchings of an arrow pointing in the direction of the magnetic field. The symbol $\bigodot$ means that the field is pointing towards you (just as the arrow would be, if you were looking at the point). The symbol $\bigotimes$ means that the field is pointing away from you (just as the arrow would be, if you were looking at the fletching).

## Questions

1. What force is exerted by a 1T magnetic field on an electron (of charge -1.6 x 10−19C) moving at 5% of the speed of light (3 x 108 ms−1)?

2. What force is exerted by a 5mT magnetic field on a 20 cm wire with resistance 1μΩ attached to a 9V battery?

3. The following diagram shows a positive charge moving through a magnetic field. Draw an arrow representing the direction of the force on the charge.

4. The following diagram shows a wire in a magnetic field. Draw an arrow representing the direction of the force on the wire.