# Can inductance be dependent on direction in plasmas?

Inductance is given by the ratio of magnetic flux to the current. For a given plasma at a particular density and temperature (which itself could be a tensor), one may define $$L = {\Phi(i) \over i},$$

where $\Phi$ is the magnetic flux and $i$ the current. But is the inductance of the plasma, $L$, the same in all directions generally? For example, in a tokamak which has both poloidal and toroidal magnetic flux, would one use the same value for $L$ in both instances? If not, what microscopic influence(s) affect the inductance to behave differently in different directions?

• It is definitely direction-dependent. One would need to calculate the electric and magnetic susceptibilities to learn more, but I am certain that the direction matters, i.e., Faraday's and Ampere's laws are vector field equations. – honeste_vivere Aug 11 '18 at 21:08
• That's what I was thinking, yet in certain texts (e.g. page 20 here), when calculating toroidal force balance, only the inductance in a certain (i.e. poloidal) direction is applied but I am unsure why this is the case. Would you happen to know why the toroidal inductance is excluded? Is it simply assumed to be constant (i.e. dL/dR = 0 for the toroidal inductance)? – Mathews24 Aug 11 '18 at 22:22
• I am guessing either a symmetry or the drifts are too slow to cause much induction. The bounce motions between the top and bottom of any arc on a given azimuth is very fast. Things that happen "slowly" compared to relevant time scales can be considered effectively static. You will recall that inductors act like low-pass filters, so static things do not really care about inductors (after the initial turning-on of a power supply, of course). – honeste_vivere Aug 12 '18 at 12:28