Doubt about a unit vector in toroidal coordinates

[Ark236]

Homework Statement

i need to determinate the poloidal vector in toroidal coordinates

Relevant Equations

in spherical coordinates

x=[R+rsin(theta)] cos(phi)
y=[R+rsin(theta)]sin(phi)
z=r cos(theta)

theta unit vector = (cos(theta)cos(phi),cos(theta)sin(phi),-sin(theta))

in toroidal coordinates

x=t sinh(eta) cos(phi)/(cosh(eta)-cos(xi))
y=t sinh(eta) sin(phi)/(cosh(eta)-cos(xi))
z= t sin(xi)/(cosh(eta)-cos(xi))

where t=sqrt(R^2-r0^2)= R0 tanh(eta0) is a positive constant.

The system considers a torus that has a wire wrapped around it, through which a current flows. In this way, a field originates in the phi direction.

The direction of current is "theta" in the spherical coordinate system but in toroidal system, in several book shows that the electrical current only have xi direction and not eta direction. This is correct?

 

[Nate810]

Yes, this is correct. The current flow in a torus only has one direction: the direction of the wire wrapped around it. This direction is usually denoted as the xi direction in toroidal coordinates.