 Magnetic power balance on an expanding circular guide

For laboratory velocity $v\hat{r}+w\hat{\theta }$ and field $B\hat{z}$, $F_{B,r}=qBw$ and $F_{B,\theta }=-qBv$. Their powers are opposite. The field-dependent guide-reaction power is $-qBvw$; the total ideal radial reaction for bead mass $m_b$ and constant expansion speed additionally contributes the ordinary centripetal term $-m_{b}v{w}^2/R$.