Finding velocity field components from electric and magnetic field components in ideal MHD

Question

How can I determine the components of the velocity field ($V_x, V_y, V_z$) using the components of the electric field $(E_x, E_y, E_z)$ and magnetic field $(B_x, B_y, B_z)$, assuming the ideal magnetohydrodynamics (MHD) equation $$\vec{E} = -\vec{V} \times \vec{B},$$ and the condition $$\vec{E} \cdot \vec{B} = 0.$$ I have attempted to estimate the determinant of this system, but it yields zero. Are there any additional constraints or methods that I may have overlooked?

Answer 1

Solving the cross product equation is a well known problem. You can prove that if $\mathbf P=\mathbf Q\times \mathbf X$ (and $\mathbf P\cdot \mathbf Q =0)$ for vectors $\mathbf P,\mathbf Q$ and unknown vector $\mathbf X$, then you have that $$\mathbf X=\frac{\mathbf P\times \mathbf Q}{|\mathbf Q|}+k \mathbf Q$$ where $k$ is a scalar that has to be determined from another equation. You are right to have a zero determinant, you can see that the solution above works for any value of $k$.

The equations of ideal magnetohydrodynamics (MHD) are actually more than 2, you are probably missing other common fluid equations like:

1. the conservation of mass $\partial_t\rho+\nabla\cdot(\rho \mathbf V)=0$
2. the conservation of momentum $\partial_t (\rho \mathbf V)+\nabla\cdot (\rho \mathbf V \otimes \mathbf V)=-\nabla P +\text{other force terms}$
3. An equation for the energy or for pressure $P$

You will also need boundary/initial conditions and fluid properties to completely determine your system.

Note also that the equation $\mathbf E\cdot \mathbf B=0$ is not always true in ideal MHD, so you should check the hypotheses you are considering.