- #1
DavideGenoa
- 153
- 5
Let us assume the validity of Ampère's circuital law[tex]\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}[/tex]where ##\mathbf{B}## is the magnetic field, ##\gamma## a closed path linking the current of intensity ##I_{\text{linked}}##.
All the derivations of the Biot-Savart law for a tridimensional distribution of current ##\mathbf{B}=\frac{\mu_0}{4\pi}\int_V \frac{\mathbf{J}\times\hat{\mathbf{r}}}{r^2}dv## that I have been able to find on line and on cartaceous resources use Dirac's ##\delta##.
Can the Biot-Savart law, at least in the case of a linear distribution of current ##\mathbf{B}=\frac{\mu_0}{4\pi}\oint\frac{Id\boldsymbol{\ell}\times\hat{\mathbf{r}}}{r^2}## or, explicitly, [tex]\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\int_a^b I\boldsymbol{\ell}'(t)\times\frac{\mathbf{x}-\boldsymbol{\ell}(t)}{\|\mathbf{x}-\boldsymbol{\ell}(t)\|^3}dt[/tex] where ##\boldsymbol{\ell}:[a,b]\to\mathbb{R}^3## is a parametrisation of a closed (or infinite) wire carrying the current ##I##, be inferred without using Dirac's ##\delta##, only by using the tools of multivariate calculus and elementary differential geometry, if we assume the validity of Ampère law (at least assuming the validity of the Gauss law for magnetism or other of the Maxwell equations)?
All the derivations of the Biot-Savart law for a tridimensional distribution of current ##\mathbf{B}=\frac{\mu_0}{4\pi}\int_V \frac{\mathbf{J}\times\hat{\mathbf{r}}}{r^2}dv## that I have been able to find on line and on cartaceous resources use Dirac's ##\delta##.
Can the Biot-Savart law, at least in the case of a linear distribution of current ##\mathbf{B}=\frac{\mu_0}{4\pi}\oint\frac{Id\boldsymbol{\ell}\times\hat{\mathbf{r}}}{r^2}## or, explicitly, [tex]\mathbf{B}(\mathbf{x})=\frac{\mu_0}{4\pi}\int_a^b I\boldsymbol{\ell}'(t)\times\frac{\mathbf{x}-\boldsymbol{\ell}(t)}{\|\mathbf{x}-\boldsymbol{\ell}(t)\|^3}dt[/tex] where ##\boldsymbol{\ell}:[a,b]\to\mathbb{R}^3## is a parametrisation of a closed (or infinite) wire carrying the current ##I##, be inferred without using Dirac's ##\delta##, only by using the tools of multivariate calculus and elementary differential geometry, if we assume the validity of Ampère law (at least assuming the validity of the Gauss law for magnetism or other of the Maxwell equations)?