Explanation of different magnetic field measurements due to frame of reference

Question

If there is an stationary infinitely long wire carrying current I, and we turn on a gaussmeter, it will record a magnetic field B.  Since I know the Biot-Savart law, I am satisfied with this measurement.

As we are measuring this, let's say we notice someone else, person B, moving in the direction of current at 0.00028 m/s, the drift velocity of electrons in a copper current. Person B sees a stationary charge in a line of infinite length. They pull out an electric field meter and measure an electric field E. They also take a look at their gaussmeter and find B=0. Person B knows about the Coulomb force, and is satisfied with their findings.

So I measure $vec{B} neq 0$ and person B measures $vec{B} = 0$, in the same space! And we are both satisfied with the different measurements.

As I think about it, the only way I can reconcile this is by saying that electric and magnetic fields are the same physical phenomenon, and we have different names for them depending on frames of reference. But if that is the case, then I wonder why light propagates as waves through magnetic and electric fields at the same time? If they were 'the same thing', why would a distinction between them be necessary to explain light propagation.

My question is what is the explanation for and what are the implications of my assumption that, while I measure $vec{B} neq 0$, the moving person measures $vec{B} = 0$.

alexvas · Accepted Answer

I'm confused what you are trying to reconcile. The magnetic field is given by the Biot-Savart Law:

$$ dvec{B} sim frac{Idvec{s}times vec{r}}{r^3} $$

where

$$ I = int vec{J}cdot dvec{A} = int rho ;vec{v}_{text{drift}}cdot dvec{A}$$

So taking the Galilean transformation $vec{v}_{text{frame}} = vec{v}_{text{drift}}$ leads to:

$$ vec{J}=rho ;vec{v}_{text{drift}} rightarrow rho ;(vec{v}_{text{drift}}-vec{v}_{text{drift}}) = 0 $$

So yes, person B observes no magnetic field. Notice that the electric field of a moving electron (or a current a long a wire) also depends on the drift velocity (see Google or Jackson for the exact relation), the electric field measured by person B should also be different from person A. Yes, there are relationships between magnetic and electric fields, and in some cases they behave similarly, but overall the two are different phenomena that are unified by the field of electromagnetism. Both are necessary to describe an electromagnetic system, such as light propogation.

Note: when approaching an E&M problem the first thing you do is select a $textbf{gauge}$, which is not unlike a reference frame, so as to fix all measurements. You choose one gauge (just like you choose one $vec{v}_{text{frame}}$) and then both magnetic and electric fields are fixed [again, emphasizing that they are related, but different].

Explanation of different magnetic field measurements due to frame of reference

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