The Law of Biot and Savart again

[wofsy]

The magnetic field of a steady current in a loop is given by the Biot and savart integral which is

1/4pi Integral[((x-y)/|x-y|^3) x dy] = B(x)

What is the corresponding formula for the vector potential?

 

[saunderson]

Although i can't decrypt the formula you have stated, i give it a try

$$\vec A(\vec r) = \frac{\mu_0 I}{4\pi} \oint \limits_{\mathcal{C}} \mathrm d\vec r^{\, \prime} \, \frac{1}{|\vec r - \vec r^{\, \prime}|}$$
​

The curve $$\mathcal{C}$$ is parameterized through $$\vec r^{ \, \, \prime}(t)$$ !

It is necessary to mention, that the curve $$\mathcal{C}$$ must be closed, otherwise the integral diverges!


Best regards...

 

[wofsy]

Although i can't decrypt the formula you have stated, i give it a try

$$\vec A(\vec r) = \frac{\mu_0 I}{4\pi} \oint \limits_{\mathcal{C}} \mathrm d\vec r^{\, \prime} \, \frac{1}{|\vec r - \vec r^{\, \prime}|}$$
​

The curve $$\mathcal{C}$$ is parameterized through $$\vec r^{ \, \, \prime}(t)$$ !

It is necessary to mention, that the curve $$\mathcal{C}$$ must be closed, otherwise the integral diverges!


Best regards...

thanks I will try to prove it works.

BTW: how do you do the math notation?

 

[wofsy]

Although i can't decrypt the formula you have stated, i give it a try

$$\vec A(\vec r) = \frac{\mu_0 I}{4\pi} \oint \limits_{\mathcal{C}} \mathrm d\vec r^{\, \prime} \, \frac{1}{|\vec r - \vec r^{\, \prime}|}$$
​

The curve $$\mathcal{C}$$ is parameterized through $$\vec r^{ \, \, \prime}(t)$$ !

It is necessary to mention, that the curve $$\mathcal{C}$$ must be closed, otherwise the integral diverges!Best regards...

Thanks that works.
What about if you have an arbitrary divergence free field defined in space minus possibly a finite number of loops?

For instance if I have two magnetic fields generated by two non-linking current loops their cross product is divergence free. If there an integral formula for the vector potential of the cross product?

Or - suppose the magnetic field is confined to the interior of a closed tube as in a magnetic filament.