I am a mathematician and have a paper which models a situation where a homogeneous magnetic field is applied to a moving electrically conducting fluid. There is a Lorentz force formula on which all the work is made:
$$\mathbf{F}= \sigma (\mathbf{E} + \mathbf{V} \times \mathbf{B}) \times \mathbf{B}.$$
But I think that the last $\times \ \mathbf{B}$ is not needed. Is this the correct formula for this case, or is the work not correct?
Let us start by summarising the governing equations of MHD. We have the reduced form of the Maxwell equations
$$\nabla \times \mathbf{B} = \mu \mathbf{J},$$ $$\nabla . \mathbf{J} = 0,$$
$$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t,$$ $$\nabla.\mathbf{B} = 0$$
and the auxillary equations
$$\mathbf{J} = \sigma(\mathbf{E} + u \times \mathbf{B}),$$ $$\mathbf{F} = \mathbf{J} \times \mathbf{B}.$$
From the last two equations you get your result, namely that
$$\mathbf{F} = \sigma(\mathbf{E} + u \times \mathbf{B}) \times \mathbf{B}.$$
Note that the above combine to give the induction equation.
I hope this helps.
