I have a couple of short questions regarding the magnetic field of a toroidal coil.
Question:
- In the attached example it is shown that the magnetic field at $\vec{r}$ due to the current element $\vec{r'}$ is in the $\hat{y}$ direction. How does this proof imply that in general the field points in the $\hat{\phi}$ direction as is stated? Is this implied from the assumption that we could do this to all points (since we took an arbitrary point $\vec{r}$) and basically sum the contributions which would then still give us a vector in the $\hat{y}$ direction?
- The definition of circumferential in the context seems to be that the field points in the $\hat{\phi}$ direction? Is the general definition of a circumferential field: "the field is perpendicular to the circumference of the cross-sectional area of the shape"?
Thanks.