I’m trying to reconcile what the math of Ampere’s Law says with what “actually happens” in a physical sense…and I’m getting very confused.
If we take two parallel wires with equal currents running through them, but in opposite directions, and place an Amperian surface around them, AL says the magnetic field will be zero. But how can this be??? Clearly I’ll feel a net magnetic force anywhere that I’m closer to one wire than the other (ie, not equidistant).
Similarly, we examined solonoids too. In the case of a solonoid, we sliced it longitudinally so as to expose the wires coming “in” and “out” of the page. In this case we examined an Ameperian surface of only half the coil, that is, the coils coming “out”. We did this, having seen the above example, because the opposite currents would yield a 0 magnetic field. But clearly, in “real life” you don’t ignore half the solonoid! The “in” part IS there, contributing its opposite current. So if the math says the field should be zero, why is it not zero???
RM Mentock
Sounds like the answer my physics prof gave me…doesn’t help me understand the physical happenings any better though. When I use the formula I get zero. When I do the experiment, there’s a field, not zero. What gives?
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I understand that the individual fields cancel outside the coil and add inside the coil. In class, our prof took an Amperean surface around the “top half” of the coil. When asked why we don’t double the non-zero answer we got, he explained that if we took an AS around both surfaces, the opposite currents would cancel and yield a zero magnetic field. Ergo, we only take a surface around half the coil.
But clearly, in real life, that “bottom half” DOES exist and DOES add to the field.
Sorry, I didn’t read your question very well. When you perform an integration around a loop the integral has a direction. When you intergrate around the “in” part you’re integrating in a different direction wrt to the B field and therefore you get the same answer as you got around the “out” part.
What do you get when you integrate x[sup]3[/sup] from -1 to +1? Same thing. x[sup]3[/sup] is only zero at one point, but the total integral is zero.
If you’re take some clever paths, you can use the Law to determine what the field is not just around the path but at each point–and it might not be zero, as you say.
The result of the math STILL seems to not agree with experimental observation. The math yields a result of zero. The field is not zero. Seems like a pretty useless law to me.
When using Gauss’ Law for electric fields and Ampere’s Law for magnetic fields, one must be careful not to oversimplify. The reason that many students don’t get this (IMHO) is that they only see magnetic fields calculated for very particular cases with particular symmetries. While the teacher undoubtedly explains the requirements for those particular situations, I think this often slips by the students.
Anyway, the confusion here arises from this: Ampere’s law does not say that the magnetic field is necessarily zero around an enclosed current that totals zero. What it says is that the line integral of the scalar product of B and ds (were ds is the differential line element along the loop) is zero around a net zero current.
In order to use the “simplified” Ampere’s law that you’ve probably seen and may be thinking of:
BL = µ[sub]0[/sub]I[sub]encl[/sub]
(where L is the length of the closed loop), two conditions have to be met:
The magnetic field must be either perpendicular or parallel to the length element ds at every point, and
The magnitude of the magnetic field must be constant when it is parallel to ds, so the scalar product with B just has magnitude B. (Of course, that scalar product is zero when the two are perpendicular).
These conditions are not met in your two-wire situation. The magnetic field has a much more complicated (though not difficult to calculate, since you know the field for one wire) shape than it does with just one wire. If your Amperian loop is simply a circle, than the scalar product of B and ds changes as you go around the loop, and hence doing that integral is much more complicated.
So the bottom line is, Ampere’s Law in this case is telling you only that that particular integral is zero; it is not telling you that the magnetic field is zero. A true, but not very useful (or, as you are seeing, easily understandable in physical terms) statement.
Philbuck, you basically just stated exactly what I was trying to get at: Why is this a useful concept then, and why do we do these symmetries if the math, admittedly, does not model what is happening?
Trigonal Planar, I think you’re confusing two different integrations you might perform to determine the magnetic field. The first way you (maybe) learned is to integrate over the currents. In this case, you either integrate over all the currents, or if you have symmetry, integrate over a fraction and use symmetry to scale the result.
The other way is useful in some cases, like the solenoid. You know that the path integral of the magnetic field around a current equals the total current flowing through the loop. You’re making the assumptions that the magnetic field is approximately zero outside the solenoid, that it’s approximately constant inside the solenoid, and that (assuming you’re integrating a rectangular loop) the contributions from the short ends of the loop don’t contribute much. (You may be assuming an infinitely long solenoid also.) With these assumptions, three of the four sides contribute zero to the magnetic path integral, so you know the magnetic field is just the total current of the solenoid divided by the length of the solenoid. In this cases, you aren’t integrating over sources, so you don’t add in contributions from the top half. The top half currents don’t go throguh the loop.
