\frac{\partial \vec{B}}{\partial t} = \vec{\nabla} \times (\vec{v} \times \vec{B}) + \frac{\eta}{4\pi} \nabla^2 \vec{B} + \frac{1}{e} \vec{\nabla} \times (\frac{\vec{\nabla} P}{n})

\frac{\partial \vec{B}}{\partial t} = \frac{1}{e} \vec{\nabla} \times (\frac{\vec{\nabla} P}{n})

Physical equation units

\frac{B}{t} = \frac{1}{e} \frac{1}{L^2} \frac{P}{n}

Computational equation

\frac{B_c}{t_c} = \alpha \frac{1}{L_c^2} \frac{P_c}{\rho_c}

Divide physical equation by computational equation

\frac{B_s}{t_s} = \frac{1}{\alpha e}\frac{1}{L_s^2}\frac{P_s}{n_s}

and then solve for \alpha and simplify

\alpha = \frac{t_sP_s}{eB_sL_s^2n_s} = \frac{\rho_s}{eB_st_sn_s}= \frac{\chi}{e t_s B_s}