Why currents?

[Quetzalcoatl]

There is something that I had never been able to understand about the way electro-magnetism was formulated mathematically. In EM we say that the charge is the source of the electrical part of the field, and that the currents are responsible for the magnetic part. But what is a current? Why define such an entity? It seems to me that the current densities can be extracted from the charge density's function. Isn't there any mathematical way to get J(x,y,z,t) given just rho(x,y,z,t)?

 

This is also relevant to aerodynamics/hydrodynamics/field theory etc...

[swansont]

What of you don't know rho(x,y,z,t)? There isn't a unique mapping between rho and J.

[Bignose]

Same thing in hydrodynamics. A gradient in density does cause a fluid flow, but isn't the only reason for a fluid flow. There is an entire branch of hydrodynamics wherein the density is assumed to be a constant, and yet there is plenty of interesting flows to study.

[proton]

There is something that I had never been able to understand about the way electro-magnetism was formulated mathematically. In EM we say that the charge is the source of the electrical part of the field, and that the currents are responsible for the magnetic part. But what is a current? Why define such an entity?

Current is defined, of course, as the flow of charge. The term is defined because it appears as the source term in the defining equations for the magentic field. In relativity the Electric and Magnetic fields are unified into one field, the Electromagnetic field. When Maxwell's equations are expressed in tensor form the source term in the equations is called the 4-current and is defined in terms of charge density and current density (current per unity area).

It seems to me that the current densities can be extracted from the charge density's function. Isn't there any mathematical way to get J(x,y,z,t) given just rho(x,y,z,t)?

No. It's possible to have more than one charge density in the presense of a given current density and vice-versa. That's one of the reasons why the source is given in terms of charge density and current density.

[Quetzalcoatl]

OK, I'm still not so sure about this.

 

A gradient in density does cause a fluid flow, but isn't the only reason for a fluid flow.

What other reason can there be for flow?

 

Do u mean that, for example, in a closed wire loop, when there is a constant density there can also be a current which leaves the density constant, and we can't tell by the density alone if there's a current and how strong it is? Since there can be just about any current as long as it's constant. Meaning, that knowing the current adds information about some kinetic energy in the charge density?

 

But, I CAN detect the activation of a current. I measure the charge density without a current. Then I activate a current, and the charges start moving in the closed wire-loop. This causes the charge density in the wire to increase, according to special relativity. By measuring the new density and calculating the difference, I can figure out the amount of current added to the loop (i.e. the speed of the charges). So if I know the history of the density, I might still be able to find the current.

 

Can u give me a good example that shows how there can be a charge density function where I can't find out the current?

 

I want to understand this thoroughly precisely because it appears in relativity, with density as time-flux and currents as space-flux.

[Bignose]

What other reason can there be for flow?

 

A gradient in pressure. A gradient in surface tension. Gravity or any other body force (i.e. the forces a fluid feels in a centrifuge). Or it may even be as simple as a moving boundary drives fluid flow. A gradient in density is only one of many reasons fluid may flow.

 

Note that my statements were about fluid dynamics, which you included in your statement "This is also relevant to aerodynamics/hydrodynamics/field theory etc... ", which just isn't right.

 

I don't know electricity and magnetism nearly as well as fluid mechanics, so hopefully someone else can come along and answer your question.

[swansont]

Can u give me a good example that shows how there can be a charge density function where I can't find out the current?

 

Can you find the current if I don't tell you the conductive properties of the material? About the presence of an external field?

[timo]

Can u give me a good example that shows how there can be a charge density function where I can't find out the current?

A constant charge density on the surface of a torus.

[proton]

What other reason can there be for flow?

That's like asking what can cause a body to move. Consider a line of charge at rest on the x-axis of the inertial frame S. There is no flow of charge in this frame. Now transform to an inertial frame S' moving in the direction of the +x axis. In S' there is a current, i.e. a flow of charge.

[Quetzalcoatl]

Originally Posted by proton

Consider a line of charge at rest on the x-axis of the inertial frame S.

That would work in principle, if the line of charge was infinite. If not, at some point the charge would stop.

 

But I think I understand my difficulty now. I used to think that for a current to exist, be it because of pressure or any other force, the cause and effect relations where: Force => Density gradient => current. But now I see I was wrong. There is another way for current to exist. If a force of the same magnitude is working on all of the fluid particles (like an electormagnetic force on some electrons in a wire), they will all flow together keeping the density constant (like proton's little example).

 

Thank u all for bearing with me.

 

PS: How do I quote ppl in my post? Does

have any parameters? I'm kinda new here.

[swansont]

PS: How do I quote ppl in my post? Does

have any parameters? I'm kinda new here.

 

Set it equal to the name, and then close the tag when you're done

 

It's all relative

becomes

 

It's all relative

 

If you use the actual quote button (lower right of the post window), there will be additional information in the quite tag, which points to the post where the original quote appears

[Severian]

In electrodynamics, the current is just as fundamental as the charge density. Look at Maxwell's equations and you will often see the current on the right hand side of the equations.

 

Actually, the current and charge densities are (almost) the same thing. Since Maxwell's equations are relativistically invariant, you can write them down in a way where you treat time in the same was as space (modulo a sign in the metric). If you do that, you will find that the four dimensional current, which now has four components, has the charge density in its time component and the normal (3D) current in its space components. A relativistic transformation mixes the various components together.

 

This is actually pretty intuitive, since if I look at my charge density from a frame which is moving (say a moving train) then relative to me, the charge density is moving - it is a current! This is exactly the same reason why an electric field can become a magnetic field in a different frame of reference. If a charge density creates an electric field, while a current creates a magnetic field, then a charge density becoming a current in a different frame is the same physics as an electric field becoming a magnetic one.

[proton]

That would work in principle, if the line of charge was infinite. If not, at some point the charge would stop.

It really makes no difference the shape or extent of the charge distribution. Charge at rest in one frame means that there is current in another frame moving relative to that frame.