Relativity in magnetism

Question

Well we have defined magnetic force as an electrostatic force in a different frame of reference, based on relativity theory.

We also know that electric field and electrostatic force have same directions

Then why magnetic field has its direction perpendicular to that of electric field?

we have defined magnetic force as an electrostatic force in a different frame of reference Does your textbook really do that? — G. Smith, May 05 '20 at 17:01
No. It’s true that under Lorentz transformations a purely electric field or a purely magnetic field can become a combination of electric and magnetic fields. But this does not mean that a magnetic field is defined as a transformed electric field. — G. Smith, May 05 '20 at 17:12
The part about Lorentz transformations, or the part about defining the magnetic field? — G. Smith, May 05 '20 at 17:22
Wikipedia: “the magnetic field ... is defined as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle” — G. Smith, May 05 '20 at 17:32
@G.Smith can you tell me whether I'm right or wrong?: There exists a current and a moving charge. From the relativity stuff we have derived Lorentz law. From Lorentz law we define a field called magnetic field which is actually because of electric force from a different frame of reference. — DatBoi, May 05 '20 at 17:49
As far as I’m concerned, that’s wrong. I am seeing a message to avoid an extended discussion, so I’m done. — G. Smith, May 05 '20 at 17:55
@G.Smith can you at least specify where I have gone wrong? That would be grateful — DatBoi, May 05 '20 at 18:00
Related : Are magnetic fields just modified relativistic electric fields?. — Frobenius, May 08 '20 at 03:48

G. Smith · Answer 1 · 2020-05-05T18:14:43.037

why magnetic field has its direction perpendicular to that of electric field?

It doesn’t. In general, at any point, the electric field can have any direction and the magnetic field can have any direction. They are perpendicular in, say, a plane electromagnetic wave, but there is no relationship between their directions in a general electromagnetic field.

we have defined magnetic force as an electrostatic force in a different frame of reference

That is not how magnetic field is defined. Magnetic field is defined by the Lorentz force law. This definition requires only one reference frame.

It is a conceptual mistake to think that every magnetic field is simply a Lorentz-transformed electric field. For example, you cannot Lorentz-transform away the magnetic field of a circular current, because there is no inertial reference frame in which all the moving charges are at rest.

ProfRob · Answer 2 · 2020-05-05T18:47:59.550

"Magnetic force is an electrostatic force in a different frame of reference."

Untrue. In relativistic electromagnetism the quantity $E^2 -c^2B^2$ is an invariant.

If you have a frame of reference where there is an electric field but no magnetic field then in another frame of reference the electric field will change, but can never disappear, since $E^2 -c^2B^2>0$.

Thus in any other frame of reference there will be a mixture of electric and magnetic fields.

As to the relative direction of those fields, it is in general arbitrary, although if there is a frame of reference where there is an electric field and the magnetic field is zero (or vice-versa), then the electric and magnetic fields will be orthogonal in all other frames of reference. That is because ${\bf E} \cdot {\bf B}$ is also an invariant. Similarly if the E- and B-fields are perpendicular in one frame of reference (like for a plane electromagnetic wave in vacuum), they will be perpendicular in all other frames of reference.

There are plenty of examples where the E-field and B-field are not perpendicular. See Why are the $\mathbf E$ and $\mathbf B$ fields of an electromagnetic wave mutually perpendicular?

Relativity in magnetism

2 Answers2