Using Eq. \ref{eq:Verdet_constant} and the chain rule,
\begin{aligned} v_{c} & = & \frac{1}{L}\frac{d\theta}{dV}\frac{dV}{dB} \\ & = & \frac{1}{L}\frac{1}{V_{0}}\frac{dV}{dB}\\ \end{aligned}where we have substituted \(V_{o}\) for \(\frac{dV}{d\theta}=V_{0}\) into Eq. \ref{eq:chainrule}.