If a point charge is at rest in a frame of reference and an observer moves on an inertial frame at a velocity $\vec{v}$ with respect to the frame of reference of the point charge, according to Biot Savart's law the observer should see a magnetic field because of the moving point charge, but from the frame of reference of the charge, there is no magnetic field because the charge is at rest. How do I make sense of this contradiction. It cannot be that there is a space shared by the two frames where one sees a magnetic field and the other doesn't

  • $\begingroup$ Physical laws look the same for different inertial frames, but physical quantities can be different (no matter Galiliean or Lorentz transformation). $\endgroup$ – Ng Chung Tak Feb 5 '18 at 14:32
  • $\begingroup$ That's why we talk nowadays about the electromagnetic field. $\endgroup$ – Andrei Geanta Feb 5 '18 at 15:06

According to the Special Theory of Relativity, the electric and magnetic fields follow simple transformation rules between inertial coordinate frames. If you have an inertial system $\bar S$ moving with velocity $v$ with its $\bar x$ -axis along the x-axis of an inertial system $S$ then magnetic field in the $\bar S$ system is related to the magnetic an electric field of the $S$ system by: $$\bar B_x=B_x$$ $$\bar B_y=\gamma(B_y+\frac{vE_z}{c^2})$$ $$\bar B_z=\gamma(B_z-\frac{vE_y}{c^2})$$ with the Lorentz factor $$\gamma=\frac{1}{1-\frac {v^2}{c^2}}$$

Thus you can see that even for a zero magnetic field in $S$ ($B_x=B_y=Bz=0$), you have a magnetic field in $\bar S$ due to an existing electric field in $S$. Similar transformations hold for the electric fields. See e.g., D.J. Griffiths, Introduction to Electrodynamics, Prentice-Hall 1999.

Thus your static electric field in $S$ is seen as a magnetic and electric field in $\bar S$. This corresponds to the simple fact, that the point charge produces a current in $\bar S$ which, according to Ampere's law, has to produce a magnetic field there.


Freecharly wrote a great answer, and I would just like to elaborate some more intuitive aspects of the magnetic field. Let's assume for the time being that both the electric field and the magnetic field are derived from the corresponding forces - the electric force and the magnetic force. Today the field is typically seen as more fundamental than the force, but nothing is lost if we assume the opposite for the time being.

We have defined the electric force and the magnetic force based on observations. We can say that the electric force is the force between relatively stationary particles, while the magnetic force is the force between charges moving relative to one-another. We can see this by considering the force on a charged particle in a magnetic field as being due to a source of the magnetic field, and the source of a magnetic field is a current i.e. moving charges.

There are small problems with this. A changing electric field is also a source of magnetic field, for instance. But we resolve this for now by considering that the source of the electric field is an electric charge. While the electric field can be generated in turn by a magnetic field, we can always follow this reasoning back to a source charge. Further investigation is warranted, but will not aid our reasoning substantially. Thus we define the magnetic force as the force between charges moving relative to one-another.

Now, immediately you can see that the electric force and the magnetic force are closely related - They are both a force between electric charges. In addition, we find that the direction of the magnetic force on a nearby charge will change depending on the direction of the charge, but it will always have the possibility of being in the same direction as the electric force. In fact, the relationship between the electric and magnetic forces considered the same phenomenon, and it is possible to use special relativity to prove that magnetic force and electric force are one and the same.

We see that, if electric and magnetic force are the same, then the presence of an electric force implies the possibility of a magnetic force. Furthermore, the presence of an electric field implies the possibility of a magnetic field. But the magnetic field is not necessary under all conditions and circumstances, especially when considering problems in special relativity.

So we conclude that the electric and magnetic fields are actually the same phenomenon, the electromagnetic field, which will depend on the frame of reference.

Appendix: On the topic of the field vs. the force, from a classical perspective there is no way to tell whether the field or the force is more fundamental. By convention, an electromagnetic force implies an observer, a charge that actually observes the force. The electromagnetic field, on the other hand, is said to exist at all points where a force could be applied to a charge, if a charge were placed there. Are we to assume, then, that the field always exists just because the force is long-distance? No, rather the field is mathematically convenient and sometimes conceptually helpful.

With the benefit of 20th century physics, however, it has become apparent that there are ways to distinguish the existence of a field over a force. But that is a topic for another day.


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