Momentum and Energy of Field||Electromagnetism

2021-04-29 23:04 作者:湮滅的末影狐 0人讀過 | 我要投稿

//In?this part we discuss how energy and momentum is stored in EM field, making sure that momentum & energy conservation is always right.

//Throughout this article EM refers to?"Electromagnetic" or "Electromagnetism".

We know that conservation of energy?and momentum are?part of the basic rules of our world. In EM field, this should also be true. But something is different from what we learned in classical mechanics.

For the first part let's discuss energy of field. In electrostatic case we know the total energy writes

$V%3D%5Ciiint%20%5Cfrac12%5Crho%5Cphi%5Cmathrm%20d%20V$

But we tend to believe this energy is "stored" somewhere. And the best way is to store it in the whole field:?because it's the consequence of charge's interaction, and doesn't "belong" to any charge in the field. See the mathematical operation:

$%5Cnabla%5E2%5Cphi%3D-%5Crho%2F%5Cepsilon_0$

$%5Cbegin%7Baligned%7D%0A%20%20%20%20%5CRightarrow%20V%26%3D%5Ciiint-%5Cfrac%7B%5Cepsilon_0%7D2%5Cphi%5Cnabla%5E2%5Cphi%5C%2C%5Cmathrm%7Bd%7DV%5C%5C%0A%20%20%20%20%26%3D%5Cfrac%7B%5Cepsilon_0%7D%7B2%7D%20%5Cleft%5B%5Ciiint(%5Cnabla%5Cphi)%5Ccdot(%5Cnabla%5Cphi)%5Cmathrm%7Bd%7DV-%5Ciiint%5Cnabla%5Ccdot(%5Cphi%5Cnabla%5Cphi)%5Cmathrm%7Bd%7DV%20%5Cright%5D%5C%5C%0A%20%20%20%20%26%3D%5Ciiint%20%5Cfrac%7B1%7D%7B2%7D%5Cepsilon_0%20E%5E2-%5Cfrac%7B%5Cepsilon_0%7D%7B2%7D%5Ciint_S%20%5Cphi%5Cnabla%5Cphi%5C%2C%5Cmathrm%7Bd%7DS%0A%5Cend%7Baligned%7D$

where S is the boundary of the system. Of course we can set it infinitely far, thus? $%5Cphi%5Cnabla%5Cphi%5Crightarrow0$ , making the second integration?0. In the calculation above,? $%5Cnabla%20%5Ccdot%20(%5Cphi%5Cvec%20f)%3D%5Cphi(%5Cnabla%5Ccdot%20%5Cvec%20f)%2B%5Cnabla%20%5Cphi%20%5Ccdot%20%5Cvec%20f$ ?is used.

So the total electrostatic?energy of the system is the integration of? $w%3D%5Cfrac12%5Cepsilon_0E%5E2$ , which is called "energy density", in the whole space.

And perhaps you will guess the magnetic field also has a kind of "energy density", something like? $w_B%3D%5Cfrac1%7B2%5Cmu_0%7DB%5E2$ ... and That's Correct! Actually, we can write the total energy density of the field,

$w%3D%5Cfrac12%5Cepsilon_0E%5E2%2B%5Cfrac%7B1%7D%7B2%5Cmu_0%7DB%5E2$

More generally, the expression is:

$w%3D%5Cfrac12%5Cvec%20D%5Ccdot%20%5Cvec%20E%2B%5Cfrac12%20%5Cvec%20B%5Ccdot%20%5Cvec%20H$

We'll see more evidence that the expression above is right. If we try to calculate the change of energy density over time, it's

$%5Cfrac%7B%5Cmathrm%7Bd%7Dw%7D%7B%5Cmathrm%7Bd%7Dt%7D%3D%5Cepsilon_0%5Cvec%7BE%7D%5Ccdot%5Cfrac%7B%5Cpartial%5Cvec%7BE%7D%7D%7B%5Cpartial%20t%7D%2B%5Cfrac%7B1%7D%7B%5Cmu_0%7D%5Cvec%7BB%7D%5Ccdot%20%5Cfrac%7B%5Cpartial%5Cvec%7BB%7D%7D%7B%5Cpartial%20t%7D$

and because? $%5Cnabla%5Ctimes%20%5Cvec%7BE%7D%3D-%5Cfrac%7B%5Cpartial%5Cvec%7BB%7D%7D%7B%5Cpartial%20t%7D%2C%5C%3B%5Cnabla%5Ctimes%5Cvec%20H%3D%5Cvec%7Bj%7D%2B%5Cepsilon_0%20%5Cfrac%7B%5Cpartial%5Cvec%7BE%7D%7D%7B%5Cpartial%20t%7D$ , we can find?

$%5Cbegin%7Baligned%7D%0A%20%20%20%20%5Cfrac%7B%5Cmathrm%7Bd%7Dw%7D%7B%5Cmathrm%7Bd%7Dt%7D%26%3D%5Cvec%7BE%7D%5Ccdot(%5Cnabla%5Ctimes%5Cvec%7BH%7D-%5Cvec%7Bj%7D)%2B%5Cvec%7BH%7D%5Ccdot(-%5Cnabla%5Ctimes%5Cvec%7BE%7D)%5C%5C%0A%20%20%20%20%26%3D-%5Cvec%7BE%7D%5Ccdot%5Cvec%7Bj%7D-%5Cnabla%5Ccdot(%5Cvec%7BE%7D%5Ctimes%5Cvec%7BH%7D)%0A%5Cend%7Baligned%7D$

because $%20%5Cnabla%5Ccdot(%5Cvec%7Bf%7D%5Ctimes%5Cvec%7Bg%7D)%3D(%5Cnabla%5Ctimes%5Cvec%7Bf%7D)%5Ccdot%5Cvec%7Bg%7D-(%5Cnabla%5Ctimes%5Cvec%7Bg%7D)%5Ccdot%5Cvec%7Bf%7D%20.$

Now we admit that the energy is stored in field, so when?EM force do work?on charge, the energy of field will lose. Force on charge from EM field per unit volume is

$%5Cvec%20f%20%3D%20%5Crho%5Cvec%20E%20%2B%20%5Cvec%20j%20%5Ctimes%20%5Cvec%20B$

so the power from field per unit volume is?

$%5Cvec%20f%20%5Ccdot%20%5Cvec%20v%3D%5Crho%5Cvec%20v%20%5Ccdot%20%5Cvec%20E%20%3D%20%5Cvec%20E%20%5Ccdot%20%5Cvec%20j$

So now we know? $%5Cvec%20E%20%5Ccdot%20%5Cvec%20j$ represents energy loss of the field. And because of energy conservation, the following equantion is always right:

Energy change = -?Energy loss -?Energy transferred.

And if you've come across the concept called "flow", you may?notice at once that?energy transferred to other places per unit time is the divergence of "energy flow", which is

$%5Cvec%20S%20%3D%20%5Cvec%20E%20%5Ctimes%20%5Cvec%20H$

here? $%5Cvec%20S$ is the energy flow in field.

And for the second part we'll begin with a paradox:

A superconductive ring with permant current is placed on a?plate, which has charge? $Q$ on its edge. The radius of the plate is? $r$ , and the mangetic momentum of the ring is? $%5Cvec%20m$ .

Then we heat it up, making the ring no longer superconductive,?thus the current will reduce to 0 because of resistance. Because there do exist initial magnetic field, and now it disappears, induced electric field will appear.?We can easily find it's giving the plate a torque. So at last the plate ends up spinning, carrying angular momentum. So why the system has angular momentum at last when there?seems no initial momentum? Is the conservation of angular momentum proved wrong?

We have two ways to?explain the paradox. First is to simply deny conservation of angular momentum in EM cases; and the second way is to get angular momentum from the field, and keep momentum conservation right. And here of course we will explain in the second way.

when we see the equation

$%5Cvec%20F%20%3D%20q(%5Cvec%20E%20%2B%20%5Cvec%20v%20%5Ctimes%20%5Cvec%20B)$

which describes the EM force on a charge, there's no information of?which object is the exact source of?this force. Maybe you can say there's other charges that generate complicated EM field, so they are the origin of EM force. But there's also a reasonable explain that the force is simply from the field.?Like the?way we store energy in field, the consequence of?EM force is momentum transfer between real objects and EM field.

Back to the differential form of EM force:

$%5Cvec%20f%20%3D%20%5Crho%5Cvec%20E%20%2B%20%5Cvec%20j%20%5Ctimes%20%5Cvec%20B%3D-%5Cepsilon_0%5Cvec%20E(%5Cnabla%20%5Ccdot%20%5Cvec%20E)%2B(%5Cnabla%20%5Ctimes%20%5Cvec%20H%20-%20%5Cepsilon_0%20%5Cfrac%7B%5Cpartial%20%5Cvec%20E%7D%7B%5Cpartial%20t%7D)%5Ctimes%20%5Cvec%20B$

Once again, we write down momentum conservation:

Momentum change = - momentum loss - momentum transfer.

This time force on charge represent momentum loss of the field, so there should be?a?part representing momentum change and another representing "momentum flow".

Here we must pay attention that, unlike energy, momentum is a vector, so its "density" should be a vector, and its "flow" should be a? 2nd-order tensor, which has nine components, acting as a 3*3 matrix. So the calculation will be kind of hard, and here I'll give the conclusion directly: The momentum density is

$%5Cvec%20g%20%3D%20%5Cepsilon_0%5Cvec%20E%20%5Ctimes%20%5Cvec%20B$

which has the same direction with energy flow of field.

Back to the paradox we mentioned, now we can explain properly that angular momentum is still conservative, it only transfers from field to real objects. If you try to find the momentum density, you will see it forming loops, indicating existence of angular momentum. When an EM field forms like that, the system just have angular momentum initially. When we remove the magnetic field, angular momentum in field will return.

We also need to notice that the momentum belongs to neither? $%5Cvec%20E$ or? $%5Cvec%20B$ field; it's the interaction between them that brings momentum into EM field. For example in this paradox it's the action we push charge? $Q$ from very far place to the plate that input angular momentum to the system (Because the charge can't avoid?Lorentz force on its path).