Expansion formula for the magnetic field of a periodically deformed circular current loop
نویسندگان
چکیده
A method is derived to obtain an expansion formula for the magnetic field $\boldsymbol{B}(\boldsymbol{r})$ generated by a closed planar wire carrying steady electric current. The parametric equation of loop $\mathcal{R}(\phi)=R+Hf(\phi)$, with $R$ radius circle, $H \in [0,R)$ radial deformation amplitude, and $f(\phi)\in[-1,1]$ periodic function. based on replacement $1/|\boldsymbol{r} - \boldsymbol{r}'|^3$ factor infinite series in terms Gegenbauer polynomials, as well use Taylor series. This approach makes it feasible write circular contribution plus sum powers $H/R$. Analytic formulas are obtained from truncated finite expansions outside neighborhood wire. These showed be computationally less expensive than numerical integration regions where dipole approximation not enough describe properly. Illustrative examples due wires deformed harmonically developed article, obtaining exact coefficients accurate descriptions. Error estimates calculated identify $\mathbb{R}^3$ analytical perform well. Finally, first-order studied generic even functions $f(\phi)$. Keywords: Magnetic Field, Polynomials, Arbitrary current loop, Biot-Savart law.
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ژورنال
عنوان ژورنال: Physica Scripta
سال: 2021
ISSN: ['1402-4896', '0031-8949']
DOI: https://doi.org/10.1088/1402-4896/ac1a4e