This seems to be physics question, but the problem just concerns math.
Preface
If one wants to calculate the permeance $P$ of a rectangular bar:
it is an easy task:
$$P = \frac{\mu a b}{L} ~~~~ \rightarrow ~~~~ P\propto ab ~~~~and~~~~ P\propto\frac{1}{L}$$
where $\mu$ is the material constant. (Permeability)
Question
But my geometry is a torus with just a quarter of its circular cross section and the field $V$ passes through it parallel to the circumference of the (full) cross section:
How can I calculate the permeance of this geometry, when there are the same proportional relations as above?
Attempted solution
I divide my geometry in $N$ hollow toruses with constant wall thickness $\Delta R$ and medium length element $\Delta L$, so the field passes an area of $\Delta A$:
A little piece of the radius $R$ is $\Delta R = \frac{d}{N}$. Now one can calculate:
$$\Delta P_{n} = \frac{\mu \Delta A_n}{\Delta L_n} $$
with $$ \Delta A_n = \pi \bigg( (R+(n+1) \Delta R)^2-(R+n \Delta R)^2\bigg) $$ (Consider the full torus circumference, not just a quarter as displayed)
and $$ \Delta L_n = \frac{\pi}{2} (2n+1) \frac{\Delta R}{2} $$ (but quarter cross section!)
follows:
$$P = \sum^{N-1}_{n=0} \Delta P_{n} = \mu\sum^{N-1}_{n=0} \frac{\pi(2R\Delta R+(2n+1)(\Delta R)^2)}{\frac{\pi}{2}(2n+1)(\frac{\Delta R}{2})}~~~~~~~~~~~~~~~~~~~~~~~~~$$
$$= 4\mu\sum^{N-1}_{n=0} \frac{2r\Delta R+(2n+1)(\Delta R)^2}{(2n+1)(\Delta R)} $$
$$= 4\mu\sum^{N-1}_{n=0} \Bigg( \frac{2R}{(2n+1)} + \Delta R \Bigg)~~~~~~~~~~ $$
$$= 4\mu \Bigg( d + 2R \sum^{N-1}_{n=0} \frac{1}{(2n+1)} \Bigg)~~~~~~~~~~ $$
And this series does not converge for $N\rightarrow\infty$. Which is physically seen not possible, so there must be a problem with the math. Do you see what I'm missing?