# Exception to Ampere-Maxwell circuital law

1. Apr 1, 2014

### wprince007

Is it possible for a magnetostatic field to exhibit vector circulation when no current ( free, bound or polarization ) and no time variant electric fields traverse the area defined by the closed line integral path? In other words: are there any known exceptions to Ampere-Maxwell circuital law? In its integral form and in the absence of any changing electric fields, this law states that the line integral of H around any closed path equals the current enclosed within the path, where H is the magnetic field intensity in A/m and the current is in A. I have uncovered no such exceptions; does one exist?

2. Apr 2, 2014

3. Apr 2, 2014

### wprince007

Derivation

Given the fact that we are dealing with a magnetostatic field, we can assume there are no changing electric fields to consider and that our field is being created by a steady current, so the use of Ampere's original circuital law ( unmodified by Maxwell ) is appropriate:

$\oint$B$\cdot$dl0I, or, more simply:

$\oint$H$\cdot$dl=I,

where we take the line integral of the magnetic field intensity, H ( in A/m ) around a closed path to obtain I, the current in A enclosed by the path.

We can derive Ampere's circuital law by first using the Biot-Savart law to determine the strength of the magnetic field at a distance r from a long straight current carrying conductor(CCC), and we find that:

H=I/(2∏r)

so we can easily see that if I=0, then H=0.

Now we choose a path around the CCC that will be easy to use as a closed integral path: a circle with all of its points at a distance r from the CCC ( it is centered on the CCC ). The circumference of the path is 2∏r. So:

$\oint$H$\cdot$dl=I/(2∏r)*(2∏r)=I

So if the enclosed current is zero, the line integral of H must also be zero and there cannot be any exceptions to Ampere's circuital law, at least by this simplified and highly specific mathematical reckoning ( although this result can be demonstrated for any closed path around the CCC ) .

But: have there ever been any experimental observations that contradict Ampere's circuital law?

4. Apr 2, 2014

### UltrafastPED

If there had been exceptions found - it would be important news. It would mean that Maxwell's equations are incorrect.

Nope ... I know of no contraindications to Maxwell's equations at the macroscopic level.

5. Apr 2, 2014

No, none.

6. Apr 2, 2014

### wprince007

In response to UltrafastPED, whose latest third reply was emailed to me but for some reason was not posted:

Yes: a closed curve line integral in a conservative vector field will always yield a zero result. The interesting thing about the magnetostatic field produced solely by a steady current is that if the current is not enclosed within the chosen integration path, that portion of the field is conservative and hence has a zero line integral. If, on the other hand, any current traverses the area defined by the chosen closed path, then that portion of the field is non-conservative and the integral is no longer zero but becomes proportional to the enclosed current.

I don't really expect an exception to the Ampere-Maxwell circuital law, but we have seen Ampere's original circuital law eventually modified in three ways: once in a major way with Maxwell's inclusion of the changing electric field term, and twice in rather trivial ways with the usually implied inclusions of bound currents and polarization currents in the I or J terms, which, as far as I know, were not considered by Ampere in his original formulations. All the literature I have access to upholds this rigorously and extensively tested law of electromagnetics. My query was mainly directed at uncovering any relatively recent laboratory observations that may have called into question any aspect of the Ampere-Maxwell circuital law in its present form. I know that such questions may seem impudent bordering on ignorant, but the reason that this law and laws like it are so well established is that they have been so ruthlessly examined and tested over the years. Thanks again.

7. Apr 3, 2014

### UltrafastPED

I somehow pressed the wrong button. :)

8. May 26, 2015

### Thiago Peixoto

And what if you have a magnetic material that is NOT a linear one? The Maxwell-Ampère law is valid yet? (in this case, H is not parallel to B, you have a TENSOR magnetic susceptibility wich can depend on B - that's why it is called nonlinear)

9. May 26, 2015

### wprince007

That's an interesting observation. Assume we have a mass of ferromagnetic material with a specific degree of magnetization and there are no other magnetic field sources or changing electric fields present. Then a closed curve integral in the vicinity of the material but not passing through the material would have a value of zero, but if the closed curve followed the induction lines through the material then it would have a non-zero integral proportionate to the amount of bound current it was enclosing. But that is for flux density B, not for field intensity H. Because H exists in a generally, but not exactly, opposite direction within the material (but not outside the material), calculating the circulation becomes much more problematic with H. But the original query of this thread asked if it was possible to obtain vector circulation in a magnetic field without any free, bound, polarization or displacement currents traversing the circulation path, so that precludes the path crossing through any magnetized material.

10. May 29, 2015

### tech99

I think it still works for AC if we allow for the propagation time in the space.

11. May 29, 2015

### vanhees71

The Maxwell-Ampere Law, one of Maxwell's equations, in its local form is one of the basic equations of physics (in Heaviside-Lorentz units),
$$\vec{\nabla} \times \vec{B} +\frac{1}{c} \partial_t \vec{E}=\vec{j}.$$
There has been no violation of Maxwell's equations found. To the contrary, in its quantum form, they are among the best settled Laws of Nature ever found (agreement between experiment and theory of up to 16 digits of precision).

The magnetization of ferromagnets are a quantum effect and due to the spontaneous breaking of rotational symmetry due to socalled exchange forces, which make a state, where a macroscopic part of the electron spins are aligned to one direction an energetically favorable state compared to the one where these spins are randomly oriented. Macroscopically the magnetization can be described by a current density, which has to be taken into account as source on the right-hand side in the above written Maxwell-Ampere Law:
$$\vec{j}_{\text{mag}}=c \vec{\nabla} \times \vec{M},$$
where $M$ is the magnetization of the material under consideration.

12. May 29, 2015

### wprince007

Thank you, vanhees71. This answers my original query most precisely, which, paraphrased, asked: are there any known exceptions to the Ampere-Maxwell circuital law. Specifically I wanted to know if anyone had ever observed vector circulation in a magnetic field in the absence of any free current, so-called bound currents or polarization currents, or time variant electric fields which traverse the area defined by the circulation integral path. And thanks for the degree of precision of agreement between experiment and theory. That makes the validity of this law more emphatic.