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.(adsbygoogle = window.adsbygoogle || []).push({});

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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Magnetic field by infinite wire: convergence of integral

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads for Magnetic field infinite |
---|

A Local field problem |

I Gluing lemma / infinite corollary |

**Physics Forums | Science Articles, Homework Help, Discussion**