Ampere's Law and partial-circular wires.

[mgrantbaker]

Why can't I use Ampere's law to compute the magnetic field at the center (P) of a semi-circular wire?

If I calculate B at P due to d_theta, and then, using superposition, integrate from 0 to pi,
the result is B=uI/2R.
Biot-Savarte law gives the correct answer of B=uI/4R.

 

[Doc Al]

If I calculate B at P due to d_theta, and then, using superposition, integrate from 0 to pi,
the result is B=uI/2R.

What did you integrate to get this?

 

[pam]

Ampere's law needs axial symmetry to calulate B.
Your wire doesn't have this symmetry.

 

[mgrantbaker]

What did you integrate to get this?

Let me try to better explain my thinking.

Break the wire into infinitesmially short segments length d_theta, so that each segment approximates a straight wire with a radially symmetric magnetic field.

Calculate the magnetic field dB, as resulting only from segment d_theta, using Ampere's.
dB=uI/2(pi)R, where R is the radius of the semi-circle.

By superposition, the total magnetic field B at the center of the semi-circle, would then be the sum of all the d_theta segments, i.e., the integral of uI/2(pi)R over theta.

Digging through some other books, I see that Ampere's only applies to infinite setups. So new question: What prevents it (physically or mathematically)from being used in situations like this?

 

[Paulanddiw]

I'm wondering if you counted the magnetic field contribution from the wires that supply current to you loop.

 

[Doc Al]

Let me try to better explain my thinking.

Break the wire into infinitesmially short segments length d_theta, so that each segment approximates a straight wire with a radially symmetric magnetic field.

Calculate the magnetic field dB, as resulting only from segment d_theta, using Ampere's.
dB=uI/2(pi)R, where R is the radius of the semi-circle.

There's the problem. That's not the field dB from segment d_theta; that's the field from an infinite current.

By superposition, the total magnetic field B at the center of the semi-circle, would then be the sum of all the d_theta segments, i.e., the integral of uI/2(pi)R over theta.

Digging through some other books, I see that Ampere's only applies to infinite setups. So new question: What prevents it (physically or mathematically)from being used in situations like this?

What Ampere's law requires in order to use it to solve for the field is some kind of symmetry to make the integration of the field around the closed loop easy to do. Ampere's law always applies, but is not always useful. If you take your loop around an infinite straight current, then you can argue--by symmetry--that the field at some distance R from the wire must be the same everywhere. If the wire were finite, this wouldn't be true near the ends. Of course a small element of current is extremely finite--so treating it just like an infinite wire will be very inaccurate! That's why you use the Biot-Savart law to give you the field from an element of current.

If you are familiar with Gauss's law for electric fields, similar considerations apply. You can only use Gauss's law to calculate the electric field in certain situations that have simple symmetries that simplify the integral of the field over the closed Gaussian surface.

 

[jtbell]

A simple but not-quite-correct statement of Ampere's Law is that "the line integral of B around a loop is proportional to the current through the loop." A better statement would be something like this: "The line integral of B around a closed path is proportional to the current through a surface which is bounded by that path."

For a given path, e.g. a circle, we can imagine a lot of different surfaces that are bounded by it: a plane circle, a surface that "bows out" to one side, or even an irregular pouch. Any of these surfaces has to work in Ampere's Law, in principle. Practical calculations are another matter, of course.

This means that you can apply Ampere's Law only to currents that either form closed circuits (the real world) or come from and go off to infinity (the idealized limiting case in which part of the circuit is very far away).

If you try to apply Ampere's Law to a finite circuit element (part of a circuit), you can construct surfaces (bounded by the path of the line integral) which are either "pierced" by the current element, or which "miss" the current element altogether. That is, you get different results for the same integration path depending on which surface you imagine it as the boundary of, which is mathematically inconsistent.

 

[mgrantbaker]

Great. Thanks, Doc Al & jtbell. That pretty much clears it up.