Fig. 1: The geometry of the annulus under study.

We consider a nanoscale magnetic device which has the structure of a thick ring, with the dimensions of 10 nm in thickness and 55 nm radius, and vertical dimension of negligible size but finite. The current configuration, \(\overrightarrow{{{{{{\boldsymbol{J}}}}}}}(\theta ,t)\) runs along the ring, either clockwise or anticlockwise depending on the sign of j(t). The ring coordinate system is given by \(\hat{{{{{{\boldsymbol{T}}}}}}}(\theta )\), \(\hat{{{{{{\boldsymbol{R}}}}}}}(\theta )\) and \(\hat{{{{{{\boldsymbol{z}}}}}}}\). In the toy model, the thickness of the ring is assumed to be negligible, while in numerical simulations finite size effects are considered.