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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.
DaleSpam said:There is no proof for it, it is an empirically measured law.
transparent said: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.
DaleSpam said: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 said:Ampère's circuital law is the magnetic version of Gauss's law, if Gauss's law is intuitive, why isn't Ampère's?
transparent said:I mean, for all we know, the experiment might be erroneous.
Gauss's law seems pretty intuitive. I can't understand the others.
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'_theoremtransparent said: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.
The Ampere Circuital law is a fundamental law in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It is expressed mathematically as ∮B·ds = μ0I, where B is the magnetic field, ds is an infinitesimal element of the loop, μ0 is the permeability of free space, and I is the current passing through the loop.
The Ampere Circuital law can be derived from Maxwell's equations, specifically from the Ampere-Maxwell law, which states that the curl of the magnetic field is equal to the sum of the electric current density and the displacement current density. By applying Stokes' theorem, the Ampere-Maxwell law can be rewritten as the Ampere Circuital law in integral form.
The Ampere Circuital law is significant because it allows us to calculate the magnetic field around a closed loop by knowing the electric current passing through the loop. This is useful in many practical applications, such as designing electromagnets and predicting the behavior of electrical circuits. It also played a crucial role in the development of Maxwell's equations and the understanding of the relationship between electricity and magnetism.
Like any law in physics, the Ampere Circuital law has its limitations. It is only valid for steady currents and does not account for changing electric fields or time-varying currents. In these situations, the full form of Maxwell's equations must be used. Additionally, the law assumes that the magnetic field is constant along the closed loop, which may not be the case in certain scenarios.
Yes, the Ampere Circuital law can be applied to any closed loop, regardless of its shape or size. However, it is important to note that the loop must be completely closed and cannot intersect itself, as this would violate the basic principles of the law. Additionally, the loop must contain the current-carrying wire or conductor in order for the law to be applicable.