Hi all, I've been having a look at Thidé's treatment of magnetostatics in the free ebook www.plasma.uu.se/CED/Book/EMFT_Book.pdf A couple of questions, about embarrassingly simple matters that I've forgotten: 1) In equation (1.11), is it a mistake that the magnetic force between two current-carrying loops depends on on the position vector x? It looks to me as if it's supposed to be an integral over all the "two-body" forces between infinitesmal current elements in the two loops, but this surely shouldn't depend on the location of an arbitrary point on the first loop- or am I misunderstanding the meaning of this equation? 2) He defines the magnetic B-field as the current->0 limit of the magnetic force at a point, in equation (1.15). Is the right way to think about this firstly to think about the force that acts on a current element located at x, and then take the limit of that current going to zero? Thanks in advance.
1) You are correct that the force does not depend on the absolute position of the two-loop system. Listing x is just a formality--he specifies that one small loop is at x and the other is at x'. The important part is that the equation contains only the difference x-x', with no dependence on x itself. 2) I'm not following your question. Eq. (1.15) is not about force, it just shows that div B =0. This really doesn't depend on the current, following instead from the vector calculus relation that div(curl v)=0 for any vector v
Thanks for getting back to me. 1)I wasn't trying to ask about absolute position here, so perhaps I should try and make myself clearer. In electrostatics, I can locate a charge at a point x and it experiences a force that depends upon the separation of that point from another point x'; if I think about x' as being somehow fixed, then the force a charge experiences is a function of position x, so it makes sense to write F^[electrostatic](x). Now in magnetostatics, both the source of the magnetic force and the object that experience it are extended objects. The only sense in which it makes sense to talk about a loop being located at a point is if I take it to be sufficiently small that I can neglect its length. Now this sort of approximation is straightforward if I have something like a small charged ball and I approximate it by a charge at the centre. But the force law (1.11) instructs me to integrate over both loops, taking orientation into account. I can't simply take a random point on each loop to be "the location" of the loop, because if I change these points then the resulting tangent vectors to the loop will have totally different orientations, and hence their cross products will give totally different results. So I naturally want to think about two current elements at x and x', compute a two-body force between them, and integrate over both loops (parametrised as functions of x and x'). But doing that integration should totally eliminate any functional dependence on the positions of points on either curve. So what is the meaning of writing F(x)? Question 2) is related to 1). Eq. 1.15 uses the "magnetostatic force F(x)" to define the magnetic field B(x). If I integrate over the loops in F(x), and wash out the functional dependence on x in the process, how do I define B(x)? It seems to me like the way to proceed is to define a force F(x) that acts on a current-carrying element located at a point, but that isn't what Thidé does in eq 1.11 because he integrates over both loops, so I'm not sure if I'm missing something. Thanks.
1) Thide assumes that each loop is a rigid object, so his force is a single value that acts upon the rigid body. You could ignore the (x) part of F(x), except that he uses it in the next equation below. 2) Hmm, I wonder if my draft version of his book downloaded years ago is different than the current version. I see that Eq. (1.15) is div B = 0. I can't access your link because my firewall at work blocks foreign sites, so I'll assume you are referring to Eq. (1.14) which is [tex]\mathbf{F}(\mathbf{x}) = J \oint_C d\mathbf{l}\times\mathbf{B}(\mathbf{x}) .[/tex]Thide states earlier that he is considering small loops, so B from loop x' is assumed to be constant over all of loop x and vice versa.
Ah I see. The equations I'm talking about: [tex]\mathbf{F}^{ms}(\mathbf{x})=\frac{\mu_0 I I'}{4 \pi} \oint_{C} d\mathbf{l} \times \oint_{C'} \mathbf{l'} \times \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3} [/tex](1.11) [tex]d \mathbf{B}^{stat}(\mathbf{x}) \equiv \frac{\mathbf{F}^{ms}(\mathbf{x})}{I} =\frac{\mu_0}{4 \pi} d\mathbf{i}'(\mathbf{x'}) \times \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3} [/tex] (1.15) where [itex]d\mathbf{i}(\mathbf{x}')=I d\mathbf{l}' (\mathbf{x'})[/itex] I'm afraid I still don't understand your answer to 1). Suppose "small" means "small compared to the characteristic scale over which the B-field varies" (which Thide couldn't say at this point). That doesn't make a difference, because the tangent vector to the loop still turns through a complete revolution (!), so no matter how small it is, the corresponding force still pushes different elements in different directions, right?
No, it does make a difference because what happens depends on the orientation of each segment of the loop. Consider a simple case: If the loop lies in a plane that includes the separation vector x-x', then the force is zero. If the plane of the loop is normal to the separation vector, on the other hand, then every bit of wire contributes to the force. For a more general oblique angle or non-planar case, contributions are given by the equation.
But this is sort of my point; you can't approximate the loop as a point, so what's the use of calling it small? You need to integrate over the loop, so I don't understand what the "force at x" actually means, based on eq.(1.11); I could understand it if it was defined with reference to current elements, but it looks to me as if that should be integrated out.
I see your point that Thide is being imprecise (or downright sloppy). To continue with Thide, we have to make an effort to see what he's doing, sloppy though it is, and try to correct it. He's really working backwards from [a global integrated value of F between two rigid loops] to [the infinitesimal contributions dF to that integrated value]. dF, in turn, depends on the dI in loop 1 and on an integral over the currents in loop 2. (Now that I can see the equations, it's clear that we don't need to assume that B is constant across loop 1--it's given at each point by B(x).) Now the confusing part--B(x) is local to a segment of the loop, but F(x) is global to the entire loop. So he really should have said F, without the x dependence. [tex]\mathbf{F}^{ms}=\frac{\mu_0 I I'}{4 \pi} \oint_{C} d\mathbf{l} \times \oint_{C'} d\mathbf{l'} \times \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3}[/tex]BTW, I don't know what the superscript ms refers to... Then the next equation probably should be in terms of a differential dF(x) [tex]d \mathbf{B}^{stat}(\mathbf{x}) \equiv \frac{d\mathbf{F}^{ms}(\mathbf{x})}{I} =\frac{\mu_0}{4 \pi} d\mathbf{i}'(\mathbf{x'}) \times \frac{\mathbf{x}-\mathbf{x}'}{|\mathbf{x}-\mathbf{x}'|^3} [/tex] The expression is consistent this way because now dF(x) is position-dependent around loop 1. It still properly integrates to the force F on the whole loop. You might consider changing books--Griffiths is used widely and is very popular, and there are many other excellent texts that I and others here can recommend.
Okay, so you seem to be confirming the broad picture I had. Thanks for your help. I decided to look at Thide's text when I saw it recommended on 't Hooft's page of free web resources. I wanted something that reviewed the basics in a concise way and it seemed to fit the bill. I was planning on complementing it anyway with Griffiths when I didn't understand something really basic, but I was trying to understand Thide's presentation here as I quite like the logical structure- force law derived from experiment, then abstract from that the definition of the fields. Thanks again for your help.