Magnetism as a relativistic phenomena

In summary, the difference in the origin of the magnetic field in the two situations is due to the way time is considered and how it affects the charge density in the integrand.
  • #1
da_willem
599
1
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 let's 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

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

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

[tex]\frac{q}{1-\hat{r}_r \cdot \vec{v}/c}[/tex]

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?!
 
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  • #2
What is the explanation for this?The difference between the two situations lies in the way time is taken into account. In the case of the single particle, the integrand (the charge density) only depends on the position and time of the particle itself, so that the integral is effectively a function of the particle's state at the current time. On the other hand, for the current carrying wire, the integrand depends on the position and time of both the particle and the source, which means that the integral involves taking into account the motion of the particle relative to the source. This is what introduces the Lorentz contraction effect, as the particle moves faster relative to the source, the charge density appears to be Lorentz contracted. This is why the magnetic field in this case can be attributed to the Lorentz contraction of the moving charge density, while in the case of the single particle, it is due to the Doppler effect.
 
  • #3


I find this discussion on magnetism as a relativistic phenomenon very interesting. It is true that in both cases, the magnetic field is a result of the motion of charges. However, the origin of the magnetic field is different in the two situations.

In the case of a single particle, the magnetic field arises due to the geometry of the charge distribution and the effects of relativity, such as the Lorentz contraction. This can be seen in the equation for the vector potential, where the charge density is integrated at different times to account for the effects of relativity.

On the other hand, in the case of a current carrying wire, the magnetic field arises due to the motion of individual charges, as described by Griffiths. Here, the Lorentz contraction plays a role in explaining the force on charged particles stationary to the wire.

It is important to note that the two situations are not completely analogous. In the case of a single particle, we are considering a point charge, while in the case of a current carrying wire, we are considering a collection of moving charges. This difference in the charge distribution may explain the different origins of the magnetic field in the two situations.

Overall, the discussion on magnetism as a relativistic phenomenon highlights the complex nature of electromagnetism and the role of relativity in understanding its behavior. Further research and study in this area can lead to a deeper understanding of the fundamental principles of electromagnetism.
 

1. What is the connection between magnetism and relativity?

Magnetism is a relativistic phenomena because it arises from the motion of charged particles. According to special relativity, when a charged particle moves, it creates an electric field. This electric field then interacts with other charged particles, creating a magnetic field.

2. How does the theory of relativity explain magnetic forces?

The theory of relativity explains magnetic forces by showing that they are a result of the interaction between electric and magnetic fields. According to special relativity, these fields are actually different aspects of the same phenomenon, known as electromagnetic fields.

3. Can magnetic fields be affected by movement?

Yes, according to special relativity, magnetic fields can be affected by movement. This is because the strength and direction of the magnetic field depends on the relative motion of the charged particles that create it.

4. How does magnetism behave at high speeds?

At high speeds, magnetism behaves according to the principles of special relativity. This means that as an object approaches the speed of light, its magnetic field becomes stronger and more concentrated in the direction of its motion.

5. Are there any practical applications of understanding magnetism as a relativistic phenomena?

Yes, understanding magnetism as a relativistic phenomena has practical applications in many fields, including particle accelerators, nuclear engineering, and medical imaging. By understanding how magnetism behaves at high speeds, we can design and optimize these technologies for maximum efficiency and accuracy.

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