Proof of Ampere's Circuital Law: Using the Biot-Savart Law

[transparent]

Where can I find a proof for the Ampere Circuital law? Wherever I look, I just find a proof for an infinitely long current carrying conductor.

 

[Dale]

There is no proof for it, it is an empirically measured law.

 

[transparent]

There is no proof for it, it is an empirically measured law.

So is it an axiom? I always thought they declared a law only after some mathematical proof. I mean, for all we know, the experiment might be erroneous.

Edit:Were all Maxwell's equations experimentally determined? Gauss's law seems pretty intuitive. I can't understand the others.

 

[Dale]

There is no way to mathematically prove how the universe works. It simply isn't possible. All of Maxwells equations were obtained through experiment, as were all other physical theories.

 

[Astrum]

So is it an axiom? I always thought they declared a law only after some mathematical proof. I mean, for all we know, the experiment might be erroneous.

Edit:Were all Maxwell's equations experimentally determined? Gauss's law seems pretty intuitive. I can't understand the others.

Ampère's circuital law is the magnetic version of Gauss's law, if Gauss's law is intuitive, why isn't Ampère's?

 

[WannabeNewton]

You're confusing mathematical theorems and mathematical proofs with physical equations and empirical proof. The two are vastly different. That being said, you can more or less derive Maxwell's equations using special relativity: http://physics.stackexchange.com/qu...ived-from-coulombs-law-and-special-relativity

 

[transparent]

There is no way to mathematically prove how the universe works. It simply isn't possible. All of Maxwells equations were obtained through experiment, as were all other physical theories.

I suppose you are right. I always imagined that the basic axioms of the universe would be much simpler than this.

Ampère's circuital law is the magnetic version of Gauss's law, if Gauss's law is intuitive, why isn't Ampère's?

Gauss's law is simply based on the fact that if any curve enters/exits a closed Gaussian surface, it must exit/enter it as well, as long as it does not have an end/origin bounded by the closed surface. Ampere's circuital law is completely different.

 

[AlephZero]

I mean, for all we know, the experiment might be erroneous.

Sure. And if you figure out how to do an experiment that shows Maxwell's laws are erroneous (in their relativistic formulation), you would probably be in line for a Nobel prize. That's how science works!

Gauss's law seems pretty intuitive. I can't understand the others.

Sorry, but there is no law of Physics which says "the universe is so simple that everybody can understand it". Keep trying - you will probably get there eventually.

 

[BruceW]

Gauss's law is simply based on the fact that if any curve enters/exits a closed Gaussian surface, it must exit/enter it as well, as long as it does not have an end/origin bounded by the closed surface. Ampere's circuital law is completely different.

We are really talking about two mathematical theorems here (as WannabeNewton mentioned). The divergence theorem and the Kelvin-Stokes theorem. Now you say they are completely different, but actually, they are both special cases of the same theorem: the generalised Stokes' theorem http://en.wikipedia.org/wiki/Stokes'_theorem
That may not be much help to you now if you have not learned much differential geometry. (I haven't really either). But you can sleep soundly, safe in the knowledge that there is an elegant generalisation :)

 

[mikeph]

All of Maxwell's laws were empirically derived, at least until Maxwell's correction to Ampere's law; I'm not certain on the history of that part.

These are of course the integral forms.

To get to the differential forms you do need the vector calculus identities in addition to a continuum approximation.

 

[BruceW]

I wouldn't really say that. In my opinion, with any physical law, there is never 100% empirical reasoning behind it. But I would agree that some physical laws are more empirically motivated than others, historically speaking.

 

[vanhees71]

Well, I don't know what to prove, if I haven't given the basis (axioms) I'm allowed to use. It depends on the point of view what you consider the fundamental laws of nature. In the case of classical electrodynamics the fundamental laws are Maxwell's equations in the vacuum, and there Ampere's circuital law is the integral version of the Maxwell-Ampere equation (in Heaviside-Lorentz units),
$$\vec{\nabla} \times \vec{B}-\frac{1}{c} \partial_t \vec{E}=\vec{j},$$
for the case of stationary, i.e., time-independent fields. Then the local form reads
$$\vec{\nabla} \times \vec{B}=\vec{j},$$
and you get the circuital law by integrating over a surface, using Stokes integral theorem,
$$\int_{\partial F} \mathrm{d} \vec{x} \cdot \vec{B}=\int_{F} \mathrm{d}^2 \vec{F} \cdot \vec{j}=i_{F}.$$
Here, the convention of the orientation is the standard one, i.e., given the orientation of $F$, i.e., the direction of the area-element vectors $\mathrm{d}^2 \vec{F}$, the direction of the tangent vectors $\mathrm{d} \vec{x}$ of its boundary curve $\partial F$ is according to the right-hand rule. The direction of the area-element vector determines the sign of the current through the area, $i_F$.

 

[Javiershadow1]

Of course it can be proved, as long as you suppose it's true the Biot-Savart law. Look:
Law of continuity: ∂ρ/∂t + ∇⋅j=0

∇xB=∇x(∇xA)=∇(∇⋅A)-∇²(A)=∇(μ/4π*∫∇⋅(j'/(r-r'))dv') - μ/4π*∫∇²(j'/(r-r'))dv'=∇(-μ/4π*∫j'⋅∇'(1/(r-r'))dv') + μj(r)=
=-∇(μ/4π*∫(∇'⋅(j'/(r-r')) -∇'⋅j'/(r-r'))dv') + μj=-∇(μ/4π*∫(∇'⋅(j'/(r-r')) +∂ρ'/∂t/(r-r')dv') - μj=-∇(μ/4π(∫j'/(r-r')⋅dS' +∫∂ρ'/∂t/(r-r')dv')) + μj=
(Now we assume that the body is closed, so there is no current in the surface. Then, the integral over the surface is 0(j'⋅dS'=0))
=-∇(μ/4π*∂/∂t(∫ρ'/(r-r')dv'))+ μj=με∂/∂t(-∇(1/4πε*∫ρ'/(r-r')dv')) + μj=με∂/∂t(-∇Φ) + μj=μj + με∂E/∂t→∇xB=μj + με∂E/∂t

As you can see, it can be proved. I can't believe that some teachers see it as an experimental poof instead of trying to prove it.