FORCES ON MOVING CHARGES

Why is this important to XMM?

     When a net force acts on a particle acceleration occurs. If such a particle happens to be charged then that acceleration will result in the production of electromagnetic radiation.. When charges are moving in a magnetic field their motion results in a force which then accelerates them. Much of the radiation we see from the Universe is blackbody in type, that is to say, it is radiation emitted from bodies which are in reasonable thermal equilibrium such as stars.

IMAGE - A false colour image of the sun and an actual spectrum.

The spectrum of the sun shows it to be a blackbody. The irregularities in the curve are absorption peaks from the helium and hydrogen in the Sun.

    However another type of radiation that we see is called synchrotron radiation which is certainly not produced by matter in equilibrium; in fact it is produced by charges moving close to the speed of light in magnetic fields and which are thus being accelerated. Such radiation is very indicative of energetic events taking place and therefore of something exciting happening which is likely to produce some significant change that we may be able to detect. Celestial objects that emit synchrotron radiation are some of the most exotic in the Universe.

IMAGE - An image of a pulsar at the heart of the Crab Nebula.

Pulsars are incredibly energetic objects and as a consequence are not blackbodies. Instead, they emit synchrotron radiation.

     Because of its extreme sensitivity XMM will be detecting very many sources not observed hitherto, and many of these will be of this exotic kind emitting synchrotron radiation.

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What kind of force are we talking about?

     When we talk about forces on moving charges we are, in fact, talking about forces acting on charges which are moving in a magnetic field; this is known as the Lorentz force. As was discussed under the topic of Forces, E-Fields & B-Fields, fields are those regions in which forces act. Magnetic fields are therefore regions in which magnetic forces exist, though the magnetic force is not one of the four fundamental forces in nature (in that it requires its own special particle). It is enshrined in the electromagnetic force and results from the motion of electric charges.

IMAGE - Albert Einstein.

Einstein revolutionised our understanding of the Universe. His ideas on relativity showed that magnetic force is a manifestation of the electric force.

     We can see therefore that what we are strictly talking about under this topic is the forces between moving charges; the force on a moving charge in a field which is itself produced by another moving charge or charges It is a force which is quite distinct from the electrostatic force which exists between charges whether they are moving or stationary

     The magnetic force is different in another important aspect. Normally, with a given, static field, energy can be put into the field and extracted from it. A mass pushed up a hill acquires gravitational energy in the form of potential energy; a positive charge moving under the influence of an electric field gains kinetic energy. This is not the case in a magnetic field.

How is the force represented mathematically?

     The precise formulation of the Lorentz force is by the vector equation

IMAGE - The Lorentz force.

The direction of the force is always perpendicular to the velocity and the magnetic field.

     What this means is that the force, the velocity and the field have a direction as well as a magnitude, but more importantly it signifies that it is only the components of v and B which are at right angles to each other that contribute towards the magnitude of the force. Further, the direction of this force is perpendicular to the plane defined by the direction of v and B.

     Because the charge q can be either positive or negative the direction of force is opposite in each case.

     The magnitude of the Lorentz force is

     where q is the angle between the direction of the velocity of the charge and the direction of the magnetic field.

ANIMATION - The Lorentz force

Since F=BqvsinA where A is the angle between B and v, the magnitude of F will decrease with A

     For simplicity at this stage we are only going to consider the electron as our moving charge.