Lenght contraction of a charged wire and time dilation?

[cragar]

In my class we were doing examples with a current carrying wire and if we moved with a speed v we would see length contraction. And because of this the moving line charge would go up in value and we would see an Electric field. And then we did the same for a solenoid but he said we need to factor in time dilation because it takes longer for the electron to go around one of the loops. But my question is why don't we factor in time dilation for a current carrying straight wire? My teacher said it didn't matter, but i don't see why. I am sure its simple tho.

 

[vanhees71]

The total charge of the wire does not change under Lorentz transformations since it is a Lorentz invariant. What changes is the charge density. Together with the current densitiy it forms a four-vector,

$$j^{\mu}=\begin{pmatrix} c \rho \\ \vec{j} \end{pmatrix},$$

that obeys the continuity equation,

$$\partial_{\mu} j^{\mu}=0.$$

In terms of the flow of charges (e.g., electrons in the wire) this four-vector is given by

$$j^{\mu}=c\rho_{\text{rest}} u^{\mu},$$

where $$u^{\mu}$$ is the four-flow vector of the charged particles and $$\rho_{\text{rest}}=\rho \sqrt{1-v^2/c^2}$$, where $$\vec{v}=\vec{u}/u^0$$ denotes the three-velocity flow field as seen by an observer in the inertial frame under consideration. The square-root factor comes indeed from the length contraction of the volume, given as the hyper surface $$t=x^0/c=\text{const}$$ of this observer.

You can find all charge and current densities and the em. fields produced by them in any frame of reference, by expressing everything covariantly and doing the appropriate Lorentz transformations, but one schould be aware that also the usual constitutive equations must be formulated in covariant form, if really relativistic currents are involved!

 

[Naty1]

But my question is why don't we factor in time dilation for a current carrying straight wire? My teacher said it didn't matter, but i don't see why. I am sure its simple tho.

As noted, total charge doesn't change...it just gets foreshortened..density increases...after all, none of the wire "disappears" right?? So none of the charge (electrons) will either.

With a straight wire, length contraction and time dilation are inversely related via the Lorentz factor, gamma...and offsetting regarding current (charge FLOW)...more dense charge but slower flow.

Not so easy to visualize in a coil (solenoid)...but length contraction along the direction of velocity only effects a portion of the coil length...not the portion orthognoal to the direction of motion...onlythat portion in the direction of motion...a portion of the overall length... while time dilation affects it ALL...

 

[cragar]

But when you do the straight wire, the lenghts contraction and time dilation do not cancel .
You get a net E field from the length contraction . Or I am probbaly missing the point.

 

[Per Oni]

But when you do the straight wire, the lenghts contraction and time dilation do not cancel .
You get a net E field from the length contraction . Or I am probbaly missing the point.

Nope, your not missing a thing.

With a straight wire, length contraction and time dilation are inversely related via the Lorentz factor, gamma...and offsetting regarding current (charge FLOW)...more dense charge but slower flow.

Please compare this straight wire + current with a train and wagons rolling over a track.
I would agree with you that an observer stationary with the track sees the rolling stock contracted in length and can see the same numbers of wagons, but slower flow? You have to explain that one.

Going back to this wire even with a slower flow of electrons you still have to explain the op why there’s no resulting electrical field for an observer stationary with the wire.

A year or so ago I asked here exactly the same question and got also very dissatisfying answers. I remember one of the answers was to do with boundary conditions but that left me distinctly unsatisfied. The reason being that for every value of current a different value of boundary condition needs to be chosen.

For me the best answer so far is still the explanation given by Griffiths’ Introduction to Electrodynamics in part 12.3 In there no implicit time dilation, no slower flow, just length contraction. But yes also there is not the whole story fully explained.

 

[cragar]

i see , thanks for you answers