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Next: The Relativity of Mass Up: Relativistic Mechanics [8 lectures] Previous: The Twin Paradox

Electricity and Magnetism

Special relativity connects the phenomenon of magnetism and electricity. Magentism arises from the motion of charge. Different observers will record different magnetic fields, if they are in different inertial frames. In some cases, the magnetism may disapera in a given inertial frame. However, the total electro-magnetic force will still be the same for all observers.

Electric charge is relativistically invariant.

That is, a charge Q remains the same regardless of the inertial reference frame it is observed in.

As an example, consider two parallel conductors, carrying current in the same direction. Normally we would associate a ``magnetic'' field with the moving charges in each of these conductors, and declare that the interaction of these magnetic fields led to the force of attraction between them, as in figure 9.

Figure 9: The resultant ``magnetic field'' around two current carrying conductors carrying current in the same direction.
\includegraphics[width=0.7\textwidth]{parallel_cond.eps}

The electric current in the conductors is manifested by the flow of electrons, against a background of stationery ions. The actual effective speed of an individual electron is only about 1 mm/s. However, there are about Avogadro's number of electrons flowing per cubic centimetre of conductor. The overall relativistic effect of is therefore quite large.

The discussion is simpler if one considers an imaginary conductor where both the positive ions and the negative electrons flow in opposite directions in each conductor. From the point of view of Special Relativity, the electrons and ions in conductors $I$ and $II$ and the laboratory are all characterised by a reference frame in which they are at rest.

Whenever the electrons or ions are viewed from a reference frame other than the one in which they are at rest, then the distances between those charges will be Lorentz contracted, resulting in an apparent increase in the number of charges per unit length, and therefore an excess of charge of that type, for that section of the conductor. In particular, in the reference frame of the electrons(ions) from conductor $I$, the ions(electrons) of conductor $II$ will appear to be in excess. There will then be a Coulombic attraction between conductor $I$ and $II$. The same argument will hold when viewing conductor $I$ from conductor $II$. These ideas are illustrated in figure 10.

Figure 10: Two parallel current carrying conductors, viewed in three situations. Firstly, ignoring relativity. Secondly, viewing conductor $II$ from a reference frame fixed on an electron in conductor $I$. Finally, viewing conductor $II$ from a reference frame fixed on an ion in conductor $I$.
\includegraphics[width=0.7\textwidth]{conductor.eps}

To tidy up the arguments, two further points must be mentioned. In the laboratory frame, the conductor appears electrically neutral, as from this frame both charge types are subject to the same Lorentz contraction of the distance between successive charges. Also, each circuit as a whole is electrically neutral when observed from any inertial reference frame, as flow in one section of the circuit will always be compensated by the reverse flow in the opposite section of the circuit.


next up previous
Next: The Relativity of Mass Up: Relativistic Mechanics [8 lectures] Previous: The Twin Paradox
Simon Connell 2006-02-21