- #1
vibe3
- 46
- 1
Hi, I have an integral equation for which I'd appreciate any tips! The equation is:
[tex]
B^2(\vec{r}) = \int_{\vec{r}}^{\infty} \left[ (\vec{B}(\vec{s}) \cdot \nabla) \vec{B}(\vec{s}) \right] \cdot \vec{ds}
[/tex]
The path of integration can be chosen arbitrarily, starting at some point r, and ending at infinity (where B = 0). B can be thought of as a dipole magnetic field.
I've tried expanding the integrand into its components along and normal to the field lines, and choosing ds to be a path always normal to the field lines. This yields the following equation:
[tex]
B^2(\vec{r}) = -\int_{\vec{r}}^{\infty} \frac{B^2(\vec{s})}{R_c(\vec{s})} ds
[/tex]
where R_c is the radius of curvature of the field line at the point s. But here it seems difficult to continue without numerical assistance. Anyone have any ideas?
[tex]
B^2(\vec{r}) = \int_{\vec{r}}^{\infty} \left[ (\vec{B}(\vec{s}) \cdot \nabla) \vec{B}(\vec{s}) \right] \cdot \vec{ds}
[/tex]
The path of integration can be chosen arbitrarily, starting at some point r, and ending at infinity (where B = 0). B can be thought of as a dipole magnetic field.
I've tried expanding the integrand into its components along and normal to the field lines, and choosing ds to be a path always normal to the field lines. This yields the following equation:
[tex]
B^2(\vec{r}) = -\int_{\vec{r}}^{\infty} \frac{B^2(\vec{s})}{R_c(\vec{s})} ds
[/tex]
where R_c is the radius of curvature of the field line at the point s. But here it seems difficult to continue without numerical assistance. Anyone have any ideas?