Flux and differential length along the toroid

As for the problem shown in Figure 1, there are errors in the figure as pointed out by Bang Ming in a conversation with me few weeks back. The arrow current flow is correct but the exit wire should be at the bottom as corrected in the figure.

Figure 1

On top and the bottom the flux will be in the - $\inline \phi$ -direction, yielding

$\inline \vec{B}=-\hat{a_{\phi}}B_{\phi}$ .

As already explained in the previous blog, differential length is always positive. However the limit will be from 0 to $\inline -2\pi$ .

$\inline \vec{dl}=\hat{a_{\phi}}rd\phi$ .

Therefore, we have

$\inline \oint \vec{B} \cdot d\vec{s}=\int^{-2\pi}_0(-\hat{a_{\phi}}B_{\phi}) \cdot \hat{a_{\phi}}rd\phi=2{\pi}rB_{\phi}={\mu}NI$

Flux and differential length along the toroid

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