Magnetic Field: Conservative or Non-Conservative?

[i_island0]

I have posted this question earlier also. But i am changing my sentences a bit.
This is regarding conservative and non-conservative nature of magnetic field.

I have two points to make here:
(1) Since the work done by magnetic field is zero in any closed path, thus the magnetic force is conservative in nature.
(2) Since amperes law is saying that the integral of magnetic field in closed loop is not zero. Thus the magnetic field as such is not conservative.
Please comment!

So, if my understanding is correct, then i would conclude that the fact that the field is conservative doesn't mean that the force due to the field will also be conservative and vice-versa.

 

[Defennder]

(1) isn't true at all. It is true if you are talking about time-invariant E-fields, but certainly not B-fields. Otherwise Ampere's law is violated.

(2)This is correct. A steady state B-field is not conservative. Which is why \nabla \times \mathbf{B} \neq 0.

 

[cmos]

1) The OP is correct, according to strict definition, the magnetic force is conservative. This follows directly from the definition of a conservative field (take your pick: path independence, zero curl, etc.). Note, there is no violation of Ampere's law as the previous poster stated. This is because we are talking about the magnetic force as opposed to the magnetic field.

2) Correct, the magnetic field is conservative, but this applies to the magnetic field in both the quasi-static and time-varying cases.

Yes, this whole concept is counter-intuitive, but when one thinks about it, there is nothing simple about the Lorentz force to being with (which is the origin for the arguments above). With the electric force (or gravity for that matter), the forces are parallel to the associated field lines; this obviously is not the case for the magnetic field and magnetic force. Add on top of this that nothing in the theory says that the magnetic force should curl onto itself; in fact, the laws imply that it does not!

 

[Defennder]

cmos, thanks for clarifying the concept but I have some questions about the magnetic force. Is the following statement correct?

The B-field itself is a non-conservative field but the work done by a force induced by a magnetic field is path independent because unlike the E-field, the work done by a magnetic force is given by the line integral of q\mathbf{v} \times \mathbf{B} and not the line integral of B alone.

In any case, my apologies to the OP. I'm starting a course on E&M this semester and I'd like to clarify me doubts here.

 

[i_island0]

this all is going really confusing to me.
I am not making judgment on any statements given by both of them. But really, so far it didn't enter my head.
Cmos can please read your second point again. Because in my second point i was saying that magnetic field is not conservative.
Can some one put more light please.

Also my professor was saying that magnetic field is not conservative in nature. So, i ma bit more confused now.
Moreover books are saying that work done by magnetic field is zero so magnetic force is conservative in nature

 

[Defennder]

My understanding is that the work done by the magnetic force on a charged particle is zero, since the force is always exerted perpendicular to the direction of motion of the particle. Based on this, and the definition of a conservative force that the work done by such a force F is zero for a closed path, the magnetic force counts as conservative.

On the other hand, the closed path line integral of the magnetic field is not zero, as demanded by Ampere's law. I believe cmos meant to say for (2), that the magnetic field itself is non-conservative.

The apparent disparity between the two I believe, lies in the fact that unlike the E-field the force exerted by the magnetic field is always perpendicular and not parallel to the displacement of the charged particle.

 

[i_island0]

Thats sounding much better. Just one request, let's not bring E-field while discussing B-field.
Lets talk about one thing and get things right rather than mixing lots of things.
So, by definition, since work done by the force of magnetic field is zero all the time, i would say force of magnetic field is conservative.
At the same time amperean's law must be satisfied, so i would say that magnetic field is non-conservative by strict definition.

 

[cmos]

My apologies, there is clearly a typo in my post #3. You two seem to have it straight, but just to explicitly make my stance clear:
1) The magnetic force is conservative.
2) The magnetic field is non-conservative.

To me, it seems that the idea of a conservative field originated in physics as opposed to pure mathematics. This was probably also at a time when only forces were explicitly analyzed and before the concept of an always prevalent field emerged. So at this time, a conservative force was simply a force that exhibited no net work in a closed path. From that, the mathematicians must have then generalized the concept to all fields (in the mathematical sense). It is an artifact of that generalization which leads to our discussion today.

I am curious if my inference of the history is correct. Hopefully someone can contribute a nice reference...

 

[cmos]

Also, since the three of us seem to be in general agreement, I would like to pose a, somewhat philosophical, question: Does it even make sense to talk about the conservativeness of a force that, by definition, under any circumstance, can do no work?

 

[i_island0]

Yeah agree with you. But sometimes i have to set examination paper for my students.:D

 

[merryjman]

The problem, I think, is that you CAN'T speak about B-fields without speaking about E-fields also. Take an example: You operate a crane at the junkyard. You turn a knob, and the electromagnet turns on, and viola! A car is lifted up to the magnet! So the magnetic force did work, right?

No, magnetic forces can't do work. It was the changing electric field which did the work. You mention Ampere's Law as a source of your confusion, but notice that Ampere's Law explicitly demands that you consider electric charges and electric fields as the source of a magnetic field.

So both the force and the field are conservative. Sometimes it appears not to be conservative, but this is only if you incorrectly ignore E-fields.

 

[i_island0]

yes you are right.
induced electric field is the one causing the work to be done and not the magnetic field. Good point definitely.

 

[cmos]

The problem, I think, is that you CAN'T speak about B-fields without speaking about E-fields also. Take an example: You operate a crane at the junkyard. You turn a knob, and the electromagnet turns on, and viola! A car is lifted up to the magnet! So the magnetic force did work, right?

No, magnetic forces can't do work. It was the changing electric field which did the work. You mention Ampere's Law as a source of your confusion, but notice that Ampere's Law explicitly demands that you consider electric charges and electric fields as the source of a magnetic field.

So both the force and the field are conservative. Sometimes it appears not to be conservative, but this is only if you incorrectly ignore E-fields.

Nothing in that argument supports the magnetic field to be conservative. By sheer definition the magnetic field is non-conservative.

 

[i_island0]

yes.. cmos.. that's correct.. and there was a mistake in the point by merryjman.
I would rather put it this way.
magnetic field by definition is not conservative.
Induced electric field by definition is not conservative.
In the MAGLEV problem..magnetic field does no work.. so force due to magnetic field is not conservative. And its the non-conservative induced electric field that does the work.

 

[merryjman]

I've read some references that call the B-field conservative, and some that don't. One argument is the no-work-done argument, and another says that forces which depend on velocity cannot be conservative.

 

[cmos]

Are you sure you don't mean the magnetic force in those references? I could understand the debate in this regard (hence our entire conversation). But I can not expect any reputable reference calling the magnetic field conservative.

 

[i_island0]

obviously, magnetic field by definition is not conservative.