I Magnetic field by infinite wire: convergence of integral

[DavideGenoa]

Let $\boldsymbol{l}:\mathbb{R}\to\mathbb{R}^3$ be the piecewise smooth parametrization of an infinitely long curve $\gamma\subset\mathbb{R}^3$. Let us define $$\boldsymbol{B}(\boldsymbol{x})=\frac{\mu_0 I}{4\pi}\int_\gamma\frac{d\boldsymbol{l}\times(\boldsymbol{x}-\boldsymbol{l})}{\|\boldsymbol{x}-\boldsymbol{l}\|^3}=\frac{\mu_0 I}{4\pi}\int_{-\infty}^{+\infty}\frac{\boldsymbol{l}'(t)\times(\boldsymbol{x}-\boldsymbol{l}(t))}{\|\boldsymbol{x}-\boldsymbol{l}(t)\|^3}dt.$$A physical interpretation of the integral is that $\boldsymbol{B}$ represents the magnetic field generated by an infinitely long wire $\gamma$ carrying a current $I$, whith $\mu_0$ representing vacuum permeabilty.

Can we be sure that the integral converges in general and, if we can, how can it be proved?

I am posting here rather than in calculus because I suppose that the best way to approach the problem is by using the techniques of Lebesgue integration.

I notice that every component of the integral is the difference of two terms having the form $l_i'(t)(x_j-l_i(t))\|\boldsymbol{x}-\boldsymbol{l}(t)\|^{-3}$, and I see that $|l_i'(t)(x_j-l_i(t))|\|\boldsymbol{x}-\boldsymbol{l}(t)\|^{-3}\le |l_i'(t)||x_j-l_i(t)|^{-2}$, but the absolute value does not allow me to use the rule $l_i'(t)dt=dl_i$...

Thank you so much for any answer!

 

[Orodruin]

Can we be sure that the integral converges in general and, if we can, how can it be proved?

No, it is relatively easy to construct a counter-example. For example, consider the case when your curve just is a circular loop going round and round. A single turn of the loop will result in a finite magnetic field, but you have an infinite number of loops and the integral therefore does not converge.

 

[DavideGenoa]

No, it is relatively easy to construct a counter-example.

Then, when we state laws like Ampère's $\oint\boldsymbol{B}\cdot d\boldsymbol{x}=\mu_0 I_{\text{enclosed}}$ with $\boldsymbol{B}(\boldsymbol{x})=\frac{\mu_0 I}{4\pi}\int_\gamma\frac{d\boldsymbol{l}\times(\boldsymbol{x}-\boldsymbol{l})}{\|\boldsymbol{x}-\boldsymbol{l}\|^3}$ I suppose we impose other, stricter, assumption on $\gamma$ than its piecewise smoothness. What other conditions?
I heartily thank you and anybody wishing to add for your answer(s)!