So far we have discussed the forces between charges which are at rest with respect to each other. When charges are moving relative to each other they exert an additional force, the magnetic force. This force, as we will see later, plays a negligible role in inter atomic and intermolecular bonding. However, the effect of magnetic fields on atoms and molecules is used widely in all manner of spectroscopies, and we will need to appreciate magnetic behaviour in order to understand the interaction of light and other electromagnetic radiation with atoms and molecules.
Magnetic materials were discovered well before the electrostatic effects of rubbing amber. The magnetic iron oxide, magnetite, was mined in Asia Minor well before the birth of Christ, indeed by about 100 BC the Chinese had already discovered that such minerals would align with the north and south poles. By the beginning of the 19th Century it was known that such bar magnets would repel if like poles were facing each other and attract if the poles were of the opposite sense, and the similarities with the behaviour of electric charges had for some time interested scientists of the time.
In 1820 the Danish scientist Hans Christian Oersted performed a sequence of experiments which demonstrated that moving charge exerted a force on a permanent magnet. In the same year Ampère showed that as Newton's third law predicted, the opposite was also true, that a magnet exerted a force on a moving charge.
They made the following observations
* the force is proportional to the magnitude and sign of the charge on the particle
* the force is proportional to the velocity of the charged particle
* the force is proportional to the magnitude of the magnetic field
* the magnitude of the force depends on the relative orientation between the line of flight and the direction of the magnetic field
* and the force always acts perpendicularly to the particle's line of flight.
The last statement is the least expected, since so far we have only met forces which act in the same direction as the field. Mathematically the only way in which multiplying two vector quantities, in this case the velocity and the field, results in a vector which is mutually perpendicular, the force, is to apply the vector cross product.
F = qv x B (1)
where B is the magnetic field (the unit of magnetic field is the Tesla (T)). You should note that the field direction has been set by the requirement that F is perpendicular to v. However, it ensures that if we use a compass needle to map out the field the north pole will point in the direction of the magnetic field.
This force, is called the Lorentz force. (Lorentz was the Dutch physicist whose relativistic co-ordinate transformations were used by Einstein in his theory of special relativity. The rules of special relativity demonstrate that the magnetic force is in fact just the electrostatic force under a relativistic transformation. This interesting fact will not concern us here.)
The magnitude of the force is given by
F = Bqv sinθ (2).
The quantity θ is the angle between the magnetic field and the particle's direction of travel. So if a charge travelling in the region of a magnetic field experiences no force, we would say that by definition it is travelling parallel to a field line. The maximal force is exerted when it flies perpendicular to the field.
It is easiest to see the direction of this force in a drawing.
Figure 1. The Lorentz force on a positively charged ion.
One way to remember the relative direction of the force is by imagining the direction a right handed screw would move if it were turned through the angle θ from v to B.
2 The Mass Spectrometer
The classic use of the Lorentz force in the Chemistry Laboratory is with the mass spectrometer, figure 2.
Here an unknown molecule in the vapour phase is ionised (and dissociated) by being bombarded with high energy electrons. Some of the ion fragments emerge via a small aperture into a region of uniform electric field between two parallel flat plate conductors with a large potential difference between them. Here they are accelerated to form a beam of ions which emerge through the second aperture, after which they pass through a region of uniform magnetic field at right angles to their line of flight, before being detected.
We will first deal with the acceleration of the ionised fragments. If a voltage, V, is imposed across the conducting plates the potential energy of a singly ionised fragment, charge e, as it enters the field filled region is eV. The kinetic energy at this point is essentially zero. On emerging through the second aperture there is no longer any significant electric field and therefore the final kinetic energy must be eV. Hence the speed of the emerging ion is
(3),
where m is the fragment's mass.
Figure 1. The mass spectrometer. The B field is represented by a circle with a cross in the centre to show that the field is in to the page, i.e. an arrow viewed from the back.
The particle now enters the region of uniform magnetic field. Since B and v are mutually perpendicular the magnitude of the force on the ion is
(4).
The B field is shown going in to the page and hence the force acts in the direction shown in figure 2. Since the force is always at right angles to v and B the particle must follow a circular path. This means the Lorentz force is equal to the particle's mass times the centripetal acceleration and substituting for the speed we get
(5).
Hence the radius of the path taken by the ionised fragment is
(6).
So by recording the number of ionised fragments as a function of distance r in the detector plane we can measure the mass directly.