# Connection length

The connection length is understood to be the distance between two points, measured along a magnetic field line passing through these points.

In the fusion context, a distinction is made between closed field lines (inside the Last Closed Flux Surface) and open field lines (outside).

## Closed field lines

In a toroidally confined plasma, inside the Last Closed Flux Surface, the connection length is commonly defined as the length, measured along the magnetic field, to complete a poloidal turn.

In a circular tokamak, the poloidal circumference is *2πr*.
The connection length is $ L = 2\pi r/\sin(\alpha) $,
where $ \alpha $ is the pitch angle of the field line, namely
$ \tan(\alpha) = B_\theta/B_\phi $.
Assuming $ B_\phi \gg B_\theta $,
one has $ \sin(\alpha) \simeq \tan(\alpha) $, so that
^{[1]}

- $ L = 2 \pi r \frac{B_\phi}{B_\theta} = 2 \pi R q $

where *q* is the safety factor, approximated by
$ q = r B_\phi / R B_\theta $.

## Open field lines

Outside Last Closed Flux Surface, the connection length associated with a given point is defined as the shortest distance from that point to any material surface measured along the field line through that point.

## References

- ↑ K. Miyamoto,
*Plasma Physics and Controlled Nuclear Fusion*, Springer-Verlag (2005) ISBN 3540242171