Poloidal current in toroidal solenoid

[Mr. Rho]

Hi, I'm trying to figure out how the current density for a poloidal current in toroidal solenoid is written. I found you may define a torus by an upper conical ring $(a<r<b,\theta=\theta_1,\phi)$, a lower conical ring $(a<r<b,\theta=\theta_2,\phi)$, an inner spherical ring $(r=a,\theta_{1}<\theta<\theta_{2},\phi)$ and an outter spherical ring $(r=b,\theta_{1}<\theta<\theta_{2},\phi)$. I used Mathematica to illustrate the torus generated with this definition:

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So, the poloidal current density may be weitten:

$\mathbf{J}(\mathbf{r})=\frac{NI}{2\pi r\sin\theta}\lbrace\frac{\hat{r}}{r}[\delta(\theta-\theta_{1})-\delta(\theta-\theta_{2})][\Theta(r-a)-\Theta(r-b)]+\hat{\theta}[\delta(r-b)-\delta(r-a)][\Theta(\theta-\theta_{1})-\Theta(\theta-\theta_{2})]\rbrace$
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My question is: is that definition of a torus correct? my problem here is that this torus is not smooth, so I don't know if it is homeomorphic to the standard torus (I don't know much about Topology). Also I would like to know if there is a possible way to write a current density in spherical coordinates for a poloidal current in a standard toroidal solenoid:

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Thank you for your answers (:

 

[mfb]

my problem here is that this torus is not smooth, so I don't know if it is homeomorphic to the standard torus (I don't know much about Topology).

It is, but I don't see how this could be relevant in a physics problem.

Also I would like to know if there is a possible way to write a current density in spherical coordinates for a poloidal current in a standard toroidal solenoid:

Sure, but the equations could get messy.

 

[Mr. Rho]

It is, but I don't see how this could be relevant in a physics problem.
Sure, but the equations could get messy.

Thank you, I'm studying the multipole expansion of EM fields for such toroidal solenoid but I want to feel confortable with the current density before start to calculate things...