Homopolar disc dynamo ($L\dot{I}=(C\omega -R)I$)

Connect a homopolar generator to a solenoid producing its own axial field. Reinforcing winding and rotation give $C={\mu }_{0}N{a}^2/(2l)$ and $L\dot{I}=(C\omega -R)I$. A seed field grows for $\omega >R/C$, is marginal at equality, and decays below threshold. An identically zero initial field remains zero in this homogeneous ideal model; startup requires a seed or perturbation.

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.