Torque-controlled homopolar dynamo equilibrium ($|H|=\sqrt{2NG/({\mu }_0{a}^{2}l)}$)

For steady rotation at nonzero current, the electromagnetic braking torque is $C{I}^2$ and balances applied torque $G$. Thus $I^2=G/C$, while the circuit requires $\omega =R/C$. With $H=NI/l$, these give the displayed field and $G\omega =I^{2}R$. This is a steady energy balance, not a general nonlinear stability claim.