Do electric currents depend on the frame of reference?

[Meow12]

Suppose you have an infinite straight wire carrying electric current I. If you move in the direction of the electrons (opposite to the direction of the current) at the drift speed, would the current be zero in your new reference frame? Why or why not?

 

[Orodruin]

No, it would not be zero because, while the (conduction) electrons are standing still, the rest of the wire (including all positive charges) is moving.

That doesn’t mean current is not different. Together with the charge density, current density forms a 4-vector called the 4-current density, which also transforms accordingly. For example, the wire will appear to have a net charge density in other frames even if neutral in its rest frame.

 

[Dale]

Suppose you have an infinite straight wire carrying electric current I. If you move in the direction of the electrons (opposite to the direction of the current) at the drift speed, would the current be zero in your new reference frame? Why or why not?

So this can be figured out using the four-current density. This is the relativistic quantity that enters Maxwell's equations and transforms correctly between reference frames. In an inertial frame it is defined as $\mathbf J=(c \rho,\vec j)$ where $\rho$ is the charge density and $\vec j$ is the current density.

So, in a current carrying wire the electrons are drifting with some velocity $-\vec v$. So they have a four-current density $\mathbf J_e=(-c \rho, \vec v \rho)$ and the protons have a four-current density $\mathbf J_p = (c \rho,0)$, so the total current density is $\mathbf J = \mathbf J_e + \mathbf J_p = (0,\vec v \rho)$

Now, if we Lorentz transform to a frame where the electrons are at rest then we get $\mathbf J'_e=(-\gamma c \rho + \gamma \rho v^2/c,0)$, so there is no current due to the electrons. However, we also have $\mathbf J'_p=(\gamma c \rho, \gamma \vec v \rho)$ for a total current density $\mathbf J = (\gamma \rho v^2/c,\gamma \vec v \rho)$.

You can plot the total current as a function of reference frame. It turns out that the current density is the lowest in the reference frame where the charge density is 0. In all other frames the current density is $\gamma v \rho$ where $\gamma$ is based on the velocity of the reference frame wrt the uncharged frame rather than the drift velocity. Since $\gamma$ is always greater than 1 the current density is also always greater.

 

[vanhees71]

No, it would not be zero because, while the (conduction) electrons are standing still, the rest of the wire (including all positive charges) is moving.

That doesn’t mean current is not different. Together with the charge density, current density forms a 4-vector called the 4-current density, which also transforms accordingly. For example, the wire will appear to have a net charge density in other frames even if neutral in its rest frame.

The wire is neutral (i.e., has 0 charge density in its interior) in the rest frame of the electrons. It's an often made wrong statement to claim it's neutral in the rest frame of the ion lattice. For details, see the newest version of my writeup about it:

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

Right now, I'm still thinking about an approximate treatment of a coax wire of finite length with a voltage source on one end and a resistor at the other. The fully exact treatment of this seems to be not possible analytically, but an approximate treatment should be possible. We'll see...

 

[Dale]

The wire is neutral (i.e., has 0 charge density in its interior) in the rest frame of the electrons. It's an often made wrong statement to claim it's neutral in the rest frame of the ion lattice.

The wire can be any charge in any frame. The wire includes both the bulk charge (which behaves as you derived) and also the surface charge (which can be adjusted as desired). It is not a wrong statement, it is just a specification of boundary conditions.

 

[vanhees71]

The wire as a whole is of course electrically neutral (i.e., taking into account both the inner and the outer conductor of the coax cable in my writeup; both conductors for themselves are charged). Of course you have to take into account both the bulk charge densities in the wire as well as the surface-charge densities.

See Eqs. (45-48) in

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

What I was referring to was the bulk charge density within the conductors, which is 0 in the rest frame of the electrons rather than in the rest frame of the ion lattice.