Why is the conservation of electric charge expressed in the continuity equation?

[Tom Evans]

I’ve seen a few places, here is 1 such link:
http://physics.weber.edu/schroeder/mrr/MRRtalk.html

I have a problem with how they represent the charges in the frame where the free charge is stationary and the charges are moving in the wire. This seems wrong for 2 reasons:
1. They show additional charges (higher density), this seems to violate the conservation of charge.
2. The higher density violates the relativity. If you imagine bullets shot from a machine gun, the bullets would appear to shrink in length to a stationary observer, but NOT the space between them (matter shrinks, not space and space isn’t moving anyhow)

I say that it’s the atom (space between the electrons and nucleus) that shrinks, not the particles because even if they did shrink we would never notice. So in effect the atom becomes elliptical, being squashed in the direction of motion. Since atoms of the wire are not moving, there is no higher density of charges.

In the frame where the free charge is moving, it’s acceleration is reduced because of increase mass, in the frame where the charge is stationary and the wire is moving, it’s acceleration is reduced because the length of wire is shrinking. Both cases are magnetism.

What am I missing?

 

[Orodruin]

but NOT the space between them

This is wrong. Also, it is not so much that ”space” shrinks as it is that what ”space” and ”time” means that changes between inertial frames. If you go to the rest frame of the bullets you will indeed find that the spacing between them is larger than in the ground frame.

 

[Ibix]

This seems wrong for 2 reasons:
1. They show additional charges (higher density), this seems to violate the conservation of charge.

You can't have a current without making a circuit, and you'll find that the situation is reversed in the return wire. The total charge flowing around is the same in both cases.

2. The higher density violates the relativity. If you imagine bullets shot from a machine gun, the bullets would appear to shrink in length to a stationary observer, but NOT the space between them (matter shrinks, not space and space isn’t moving anyhow)

Distances are compressed, full stop, whether there's an object there or not. This is obvious if you imagine two rows of 1m long objects 1m apart, offset by 1m each:

 _ _ _
_ _ _ _

The ends are aligned, so must be aligned in all frames. So all lengths must contract.

 

[PeroK]

I’ve seen a few places, here is 1 such link:
http://physics.weber.edu/schroeder/mrr/MRRtalk.html

I have a problem with how they represent the charges in the frame where the free charge is stationary and the charges are moving in the wire. This seems wrong for 2 reasons:
1. They show additional charges (higher density), this seems to violate the conservation of charge.
2. The higher density violates the relativity. If you imagine bullets shot from a machine gun, the bullets would appear to shrink in length to a stationary observer, but NOT the space between them (matter shrinks, not space and space isn’t moving anyhow)

I say that it’s the atom (space between the electrons and nucleus) that shrinks, not the particles because even if they did shrink we would never notice. So in effect the atom becomes elliptical, being squashed in the direction of motion. Since atoms of the wire are not moving, there is no higher density of charges.

In the frame where the free charge is moving, it’s acceleration is reduced because of increase mass, in the frame where the charge is stationary and the wire is moving, it’s acceleration is reduced because the length of wire is shrinking. Both cases are magnetism.

What am I missing?

You're missing a lot. For example, if we have $n$ charges and you measure them contained in volume $V_1$ and I measure them contained in a volume $V_2$, then we measure different charge densities, but that is no violation of conservation of charge. Although, actually "invariance" of total charge would be more accurate. "Conservation" refers to before and after, not two different frames.

Also, SR predicts length contraction, which is the length of anything. An object or the space between objects. Objects are mostly space between atoms in any case.

 

[Tom Evans]

Also, SR predicts length contraction, which is the length of anything. An object or the space between objects. Objects are mostly space between atoms in any case.

Still have a problem with this, space between the moving objects isn’t moving, so SR doesn’t apply.
Yes there is more space between moving objects because of the length contraction.

I see what you are saying about charge density, guess my problem is how they show it.

 

[Orodruin]

Still have a problem with this, space between the moving objects isn’t moving, so SR doesn’t apply.
Yes there is more space between moving objects because of the length contraction.

Again, this is a fundamental misunderstanding of what length contraction is. It is a question of how space and time are perceived in different frames. Not about ”moving objects” becoming shorter.

 

[Ibix]

Still have a problem with this, space between the moving objects isn’t moving, so SR doesn’t apply.

As Orodruin says, this is the wrong way to think about it. Nothing is shrinking; rather you are changing your definition of how spacetime should be divided into space and time. So whether "space is moving" (a meaningless concept, by the way) isn't relevant.

In my last post I also showed you a simple thought experiment to see why your thinking must be wrong.

 

[Dale]

1. They show additional charges (higher density), this seems to violate the conservation of charge.

Charge is still conserved, meaning that it does not change over time in either frame. Furthermore, charge is invariant meaning that if you consider all the charge in the entire loop of wire, both frames will agree on the value.

Charge density is neither conserved nor invariant. So you are mistakenly looking at charge density and making statements about charge.

The higher density violates the relativity.

This is an odd claim since the density was derived from relativity. How can it violate relativity if it is derived from relativity?

 

[Tom Evans]

This is an odd claim since the density was derived from relativity. How can it violate relativity if it is derived from relativity?

They don’t mention a loop of wire, they show a segment of wire.
And it’s not derived, but it’s simply stated, and they show it like:

_______________
+ + + +
- - - - - - - - -
_______________

Maybe I’m taking the picture too literally, but that’s why I have a problem, see my link above.

 

[Orodruin]

And it’s not derived, but it’s simply stated

Even if it is "simply stated" in many explanations, it is actually directly derived from SR. It is a simple matter of Lorentz transforming the 4-current-density.

Edit: If you are not familiar with 4-vectors, it is also easily argued from length contraction.

Edit 2: If you would have an infinite straight wire with a current running through it, then indeed the total charge would be different in different inertial frames. However, in no way does this violate conservation of charge any more than the fact that energy being different in different frames violates conservation of energy.

 

[Dale]

They don’t mention a loop of wire, they show a segment of wire.

Of course, it is not necessary for their derivation, it would only serve to respond to your question which is itself based on a faulty premise. Specifically, you are looking at charge density and making incorrect conclusions about charge.

To get charge you have to integrate charge density over all space, and as you say they are only showing a segment. Thus, no conclusions can be made one way or the other regarding charge from what they wrote or drew.

And it’s not derived, but it’s simply stated,

They were very clear that the statements were derived from relativity.

Maybe I’m taking the picture too literally, but that’s why I have a problem, see my link above.

No, the problem isn’t taking the picture too literally. The picture is fine. The problem is confusing charge density with charge. You cannot make the inference about charge that you are attempting to make.

 

[A.T.]

I’ve seen a few places, here is 1 such link:
http://physics.weber.edu/schroeder/mrr/MRRtalk.html

I have a problem with how they represent the charges in the frame where the free charge is stationary and the charges are moving in the wire. This seems wrong for 2 reasons:
1. They show additional charges (higher density), this seems to violate the conservation of charge.
2. The higher density violates the relativity. If you imagine bullets shot from a machine gun, the bullets would appear to shrink in length to a stationary observer, but NOT the space between them (matter shrinks, not space and space isn’t moving anyhow)

Here is a good explanation by DrGreg:
https://www.physicsforums.com/threads/explanation-of-em-fields-using-sr.714635/page-2#post-4528480

 

[Sorcerer]

I’ve seen a few places, here is 1 such link:
http://physics.weber.edu/schroeder/mrr/MRRtalk.html

I have a problem with how they represent the charges in the frame where the free charge is stationary and the charges are moving in the wire. This seems wrong for 2 reasons:
1. They show additional charges (higher density), this seems to violate the conservation of charge.
2. The higher density violates the relativity. If you imagine bullets shot from a machine gun, the bullets would appear to shrink in length to a stationary observer, but NOT the space between them (matter shrinks, not space and space isn’t moving anyhow)

I say that it’s the atom (space between the electrons and nucleus) that shrinks, not the particles because even if they did shrink we would never notice. So in effect the atom becomes elliptical, being squashed in the direction of motion. Since atoms of the wire are not moving, there is no higher density of charges.

In the frame where the free charge is moving, it’s acceleration is reduced because of increase mass, in the frame where the charge is stationary and the wire is moving, it’s acceleration is reduced because the length of wire is shrinking. Both cases are magnetism.

What am I missing?

Hey Tom, are you aware that the derivation of length contraction doesn’t have to involve any object of length moving at all?

Here is the Lorentz transformation for x-coordinates (just coordinate transformations):

Δx’ = γ(Δx - vΔt)

And here is that transformation equation when
Δt = 0.

Δx’ = γΔxNotice anything? Replace Δx and Δx’ with L and L₀, then divide by γ.

L = L₀/γ

Tada! The length contraction formula!

You can derive it by assuming a length-less particle of light (usually people assume a beam or pulse). There is no massive object necessary.What you end up with is the Lorentz transformation equations, which are entirely about spacetime coordinates. Then you consider the special case where the ends between the x-coordinate interval are measured simultaneously in your frame, which amounts to setting delta time to zero. Out pops your length contraction formula.

In other words, length contraction MUST be more fundamental than mere objects, because it can be derrived from spacetime coordinates, not pieces of matter.

At least that seems fairly obvious to me.After all, what is a length if not a measurement difference of two spatial coordinates?

 

[vanhees71]

It's always good to keep things as simple as possible. The pedagogical overcomplication as exemplified by Purcell's (in)famous book (vol. 2 of the otherwise very nice Berkeley physics course) about electrodynamics is more confusing than helpful, in my opinion.

Electrodynamics is the paradigmatic example of a local classical relativistic field theory. Thus its most natural language is vector calculus (or rather tensor calculus) in Minkowski space and differential equations (i.e., the Maxwell equations in differential form).

The conservation of electric charge thus is most simply expressed in the manifestly covariant continuity equation
$$\partial_{\mu} j^{\mu}=0,$$
where $j^{\mu}=(c \rho,\vec{j})$ is the four-current-density field. Its components transform as components of a four-vector field, as suggested by the notation. Because of the validity of the continuity equation (and only because of that!) the total charge
$$Q=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \rho(t,\vec{x})$$
is (a) conserved, i.e., independent of time and (b) a Lorentz scalar, while both $\rho$ and $\vec{j}$ of course change when boosting from one inertial frame to another.