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I am trying to compute the volume charge density present in a toroidal conductor. For a current in the azimuthal direction, the potential inside the toroid is:

$$\Phi(\eta>\eta_{0},\xi,\Theta)=A+\sqrt{\cosh(\eta)-\cos(\xi)}\sum_{q=1}^{\infty}\sin(q\Theta)\sum_{p=0}^{\infty}\frac{2\sqrt{2}B(-1)^{q-1}(2-\delta_{0p})\beta(\frac{1}{2},p+\frac{1}{2})}{q\pi2^{p+\frac{1}{2}}(\cosh^{p+\frac{1}{2}}\eta_{0})}\cos(p\xi)$$

Applying the thin toroidal approximation, and taking the terms when $p=0$, the potential turns out to be: (for $\eta\gg\eta_{0}$; $\eta_{0}\rightarrow0$ as $\cos(\eta_{0})\rightarrow1$) $$\Phi(\eta>\eta_{0},\xi,\Theta)=A+\sqrt{\cosh(\eta)-\cos(\xi)}\sum_{q=1}^{\infty}\frac{\sin(q\Theta)(-1)^{q-1}}{q}.$$

Note: $\eta$ $\epsilon[0,\infty)$,

$\xi$ $\epsilon[-\pi,\pi]$, and

$\Theta$ $\epsilon[-\pi,\pi]$.

The electric field intensity can be computed, using: $E=-\nabla\Phi$, and there would be three electric fields in the directions of the three toroidal unit vectors. After calculating the net electric field by laws of vector addition, the net electric field turns out to be:

$$ \begin{aligned} E&=\left(\left[\left(\sum_{q=1}^{\infty}\frac{\sin(q\Theta)(-1)^{q-1}}{q}\right)\left(\frac{\sinh^{2}(\eta)-\sin^{2}(\xi)}{\cosh(\eta)-\cos(\xi)}\right)\right]^{2} \right. \\ &\hspace{4mm}\left. +\left[2(\cosh(\eta)-\cos(\xi))\left(\sum_{q=1}^{\infty}\Theta\frac{\sin(q\Theta)(-1)^{q-1}}{q}\right)\right]^{2}\right)^{\frac{1}{2}}. \\ \end{aligned} $$

Using Gauss Law, the volume charge density I see to be computed as follows: $$ \iint{EdS}=\frac{1}{\epsilon}\iiint_{V_0}\rho dV $$

$$ \rho=\frac{\epsilon}{\int{dV}}\int_{\xi=-\pi}^{\pi}\int_{\eta=0}^{\infty}|J|Ed\eta d\xi. $$

The Jacobian is: $$J=\det\left(\frac{\delta(x,y)}{\delta(\eta,\xi)}\right)=a^{2}\sin(2\Theta)\frac{(\sinh\eta)(\sinh\xi)[(\cosh\eta)(\cos\xi)-1]}{[\cosh\eta-\cos\xi]^{2}},$$

where $x=a\frac{(\sinh\eta)(\cos\Theta)}{\cosh\eta-\cos\xi}$ and $y=a\frac{(\sinh\eta)(\sin\Theta)}{\cosh\eta-\cos\xi}$.

How do I compute this integral and find the charge density? Any help is appreciated.

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