Let us assume the validity of the Biot-Savart law for a tridimensional distribution of current:$$\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\int_V \mathbf{J}(\mathbf{y})\times\frac{\mathbf{x}-\mathbf{y}}{\|\mathbf{x}-\mathbf{y}\|^3}\: \mathrm d^3 y$$
where $\mathbf{J}$ is the density of the current distributed over the region $V$. How could we then prove Ampère's law, in the form $$\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}\quad\text{or}\quad\nabla\times\mathbf{B}=\mu_0\mathbf{J}?$$ I would prefer a proof using multi-variable calculus only, not making use of Dirac's $\delta$, which is a mathematical tool, quite delicate to use, which would require much explanation for me to understand the equalities where it appears (I am a beginner, as you can read in my profile, but I just want to understand why physical laws are derived in a mathematically correct way and I am not interested in shortcuts unless knowing their mathematical proof) and whose utilisation sometimes found in textbooks causes enormous problems to me when the text does not say, for example, why it commutes integral and differentiation signs, like $\int$ and $\nabla\times$. Anyhow, I would be very grateful to anybody producing any proof, even using the $\delta$, but clearly explaining what theorems and mathematical results justify delicate steps such as those where integral (explicitly defined as Riemann, Lebesgue, symbolic signs for distributions or other) and differential operators commute. I heartily thank any answerer. Suggestions for proofs having such requisites that can be found on line or on printed books (better if not using the $\delta$) are very welcome! I heartily thank any answerer.
P.S.: A derivation of Ampère's law from the Biot-Savart one is here, but it encompasses the case of a linear distribution of current, which is different, as I thought to be blatantly obvious before seeing "duplicate" close votes.
Trials: Although I have been looking for a rigorous and understandable (by me) proof for more than a month, I have not been able to find, or to produce, one. The most common derivation of Ampère's law from the Biot-Savart one essentially uses the calculations found in Wikipedia's outline of proof, which are un-understandable to me: I know some properties of Dirac's $\delta$ like the fact that, if $(J_1,J_2,J_3)=\mathbf{J}\in C^\infty(\mathbb{R}^3)$ is compactly supported, then $$-\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{\nabla_l^2 J_i(\mathbf{l})}{\|\mathbf{l}-\mathbf{r}\|}\; d\mu_{\mathbf{l}}=J_i(\mathbf{r})=:\int\delta(\mathbf{l}-\mathbf{r})\,J_i(\mathbf{l})\,\mathrm d^3l$$ where the integral on the left is of the Lebesgue type, while the integral sign on the right is just a symbolic notation for a linear functional, but I have got tremendous problems in understanding the steps found in that outline of proof, for the reasons I explain here in details for potential answerers itending to modify it, by adding the explanations of the mathematical facts used to justify the commutations between $\iiint$ and $\nabla$ and of the exact meaning of the integral signs, to produce a more detailed and complete proof. Just to make a brief resumee of the problems I found in Wikipedia's outline of proof, as I have been requested by a commenter, I do not understand what those integral signs mean (Lebesgue integrals, symbolic signs for functionals or what else), neither do I understand what the derivative components of $\nabla\times\mathbf{B}$ are: since theorems such as Stokes' are usually applied when integrating $\nabla\times\mathbf{B}$, I would believe that they are the ordinary derivatives of elementary multivariate calculus, but then the $\delta$, which is a tool of the theory of distributions, pops up in the outline of proof, and in the theory of distributions there exist derivatives of distributions which are a very different thing, but they are taken, as far as I know, with respect to the variables written as "variables of integration" in the distribution integral notation, while, here, the outline of proof starts with $\nabla_r\times \mathbf{B}$ with $r$ , while the integral appears with $\mathrm{d}^3\mathrm{l}$...
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