- #1
DavideGenoa
- 155
- 5
I am trying to calculate the magnetic field generated by an ideal toroidal solenoid by using the integral of the Biot-Savart law. I do not intend to use Ampère's circuital law.
Let ##I## be the intensity of the current flowing in each of the ##N## loops of the solenoid, which I will consider an ideal continuous solenoid from this point.
If ##\mathbf{r}(u,v):[0,2\pi]^2\to\mathbb{R}^3##, ##\mathbf{r}(u,v)=(b+a\cos v)\cos u\mathbf\,{i}+(b+a\cos v)\sin u\,\mathbf{j}+a\sin v\mathbf\,{k}## is a parametrization of the torus, I would say that, in an "infinitesimal spire" of the ideal solenoid, generated by the rotation of ##du## radians of the circumference generating the torus, an "infinitesimal current" ##\frac{IN}{2\pi}du## flows and therefore I would think that the magnetic field at ##\mathbf{x}## could be expressed by $$\frac{\mu_0}{4\pi} \int_{0}^{2\pi}\int_0^{2\pi}\frac{IN \,\partial_v\mathbf{r}(u,v) \times(\mathbf{x}-\mathbf{r}(u,v) )}{2\pi\|\mathbf{x}-\mathbf{r}(u,v)\|^3} dudv.$$
Am I right?
I thank anybody for any answer.
Let ##I## be the intensity of the current flowing in each of the ##N## loops of the solenoid, which I will consider an ideal continuous solenoid from this point.
If ##\mathbf{r}(u,v):[0,2\pi]^2\to\mathbb{R}^3##, ##\mathbf{r}(u,v)=(b+a\cos v)\cos u\mathbf\,{i}+(b+a\cos v)\sin u\,\mathbf{j}+a\sin v\mathbf\,{k}## is a parametrization of the torus, I would say that, in an "infinitesimal spire" of the ideal solenoid, generated by the rotation of ##du## radians of the circumference generating the torus, an "infinitesimal current" ##\frac{IN}{2\pi}du## flows and therefore I would think that the magnetic field at ##\mathbf{x}## could be expressed by $$\frac{\mu_0}{4\pi} \int_{0}^{2\pi}\int_0^{2\pi}\frac{IN \,\partial_v\mathbf{r}(u,v) \times(\mathbf{x}-\mathbf{r}(u,v) )}{2\pi\|\mathbf{x}-\mathbf{r}(u,v)\|^3} dudv.$$
Am I right?
I thank anybody for any answer.