# Magnetism as a relativistic phenomena

1. May 18, 2007

### da_willem

I reread certain parts of Griffith's 'introduction to electrodynamics' out of interest in relativity in electrodynamics and was wondering the following.

In the last chapter Griffiths shows that the magnetic field of a current carrying wire can be shown to be the electric field arising due to the Lorentz contraction of the charges. He therefore describes the current as two superimposed line charges with opposite charge denisty, thus yielding no net charge. But then he lets one of the line charges move, as to simulate the current in the wire. But now the moving line charge density gets Lorentz contracted, so that the line charges no longer cancel and he shows that the resulting electric field yields a force on charged particles stationary to the wire, equal to what we would normally attribute to the magnetic force of the current.

Earlier on in the book however he does a similar thing for a single particle in calculating the Lienard-Wiechert potentials. Here the magnetic field arisis through the vector potential

$$\vec{A} = \frac{\mu_0}{4 \pi} \frac{\vec{v}}{r_r} \int \rho (\vec{r}',t_r ) dV'$$

where _r indicate retarded quantities and the primed variables are just dummy integration variables. But now something strange happens with this integral: because the charge density has to be integrated at different (retarded) times, for different parts of the configuration, this integral will not yield exactly the total charge q, but rather

$$\frac{q}{1-\hat{r}_r \cdot \vec{v}/c}$$

This is because objects look longer when they move towards you because the light of the parts furthest away left earlier. The same thing happens in building the potentials, it is a geometric effect that remains even if the size of the charge goes to zero, a point charge.

Now Griffiths comments that this has nothing whatsoever to do with Lorentz contraction, but its more reminiscent to the Doppler effect.

But what more is a current than a large collection of moving charges? How come the origin of the arising magnetic field is different in the two situations?!