Exploring the Limitations of Ampere's Law in Magnetic Field Calculations

[Nick O]

EDIT: I don't see this as a coursework question, even though I use a textbook example to set up my question. I feel that this is purely a conceptual question. If the staff disagrees, however, I am fine with this being moved to the homework forum.

I have been very perplexed these past few hours, trying to figure out why Ampere's law (at least, as I applied it) does not seem to yield the correct magnetic field at some distance from a wire of finite length.

Below is a picture showing the correct solution for one particular case using the Bio-Savart Law, and then a failed attempt using Ampere's law. The field found by Ampere's law would be correct if the wire were infinitely long, but I can't bring myself to believe that the law can only be applied to straight wires of infinite length.

It makes sense to me conceptually that the field around the wire should vary with distance from the endpoints, but no such limitations were mentioned when I was introduced to Ampere's law. I am therefore unsure of when it is valid and when it isn't. It clearly isn't valid here - why isn't it valid, and how do I know when it is or isn't valid?

 

[atyy]

I'm not sure. One thought that comes to mind is that the situation you set up doesn't conserve charge. If you set up a closed circuit, then there wouldn't be enough symmetry to apply Ampere's law in this simple form.

Ampere's law holds for any situation without a changing E or B field. If there is a changing E or B field, then one has to add a term usually called "Maxwell's correction". http://en.wikipedia.org/wiki/Maxwell's_equations

 

[WannabeNewton]

This is a very common question but a very good one at that! The short answer is that current flowing across a finite wire does not fall under the regime of magnetostatics whereas the form of Ampere's law you wrote down (without Maxwell's displacement current) falls under the regime of magnetostatics. In order to have a steady state current (which we need for magnetostatics) one must have a closed loop; current that is engendered at one point and terminated at another point is of course not an example of a steady state current. With an infinite wire we have the ends off at infinity and can consider the system a closed circuit for all intents and purposes but not so with a finite wire.

 

[Nick O]

Thank you both! I suppose I'll find the "long answer" later in the semester when we cover Maxwell's equations. In the meantime, it's nice to know that I'm not going insane.

 

[atyy]

BTW, what is the regime of validity of the Biot-Savart law? Is it also magnetostatics?

Edit: I looked up http://farside.ph.utexas.edu/teaching/em/lectures/node39.html and it seems the Biot-Savart law only applies (without approximation) to steady currents, so it also requires a closed loop. So the answer given by the Biot-Savart law for the wire segment seems to be only a partial answer. The full answer is obtained by integrating over the full loop that the segment is part of.

 

[Nick O]

That makes sense. The problem I used here is just a simplification of the one that led me to ask this question - the actual problem involved a closed, square loop, and the point was in the center. I solved it by applying the Biot-Savart law to one side, yielding the partial answer above, and multiplied the answer by four.

When I tried to rework the problem with Ampere's law, I was surprised to find that the law would not yield the correct partial answer.

 

[atyy]

When I tried to rework the problem with Ampere's law, I was surprised to find that the law would not yield the correct partial answer.

I think it would be interesting to check that the full solution obtained from the Biot-Savart law also satisfies Ampere's law (I haven't checked myself, but I think it will work).

BTW, I just want to mention, since you'll probably use Ampere's law in situations with changing currents, that there is an approximation regime called "magentoquasistatics" in which Maxwell's correction term (the "displacement current") is negelected. It's a very good approximation when the rates of change are "slow".

There's a short discussion in
http://web.mit.edu/6.013_book/www/book.html (section 1.0)
http://en.wikipedia.org/wiki/Quasistatic_approximation