# Question about conditions for Force Free Fields in plasma

#### Clear Mind

My question is about Force Free fields in the study of plasma stability (in MHD regime): Consider an isolated ideal plasma in an equilibrium state (where the effect of selfgravity is negligible), from the Navier-Stokes equation we get that:

$$\vec{\nabla} P = \frac{1}{c} \vec{J} \times \vec{B}$$

Now, if $P=const$ and $\vec{J}$ (in MHD $\vec{J}\propto\vec{\nabla}\times\vec{B}$) is parallel to $\vec{B}$, we get that $(\vec{\nabla} \times \vec{B}) \times \vec{B}=0$. Thus implies that:

$$(\vec{\nabla} \times \vec{B}) = \alpha(r) \vec{B}$$

That is the condition for a Free-Froce fields. So ... the question is, shouldn't be the curl of a vector always be orthogonal to the vector?

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#### mfb

Mentor
The cross-product is orthogonal to the two vectors, but the curl is not a proper cross-product. It can have a component along the direction of the vector. This is easy to see if you add a constant to B: its curl won't change, but you can change the direction of B arbitrarily.

• Clear Mind

#### Clear Mind

The cross-product is orthogonal to the two vectors, but the curl is not a proper cross-product. It can have a component along the direction of the vector. This is easy to see if you add a constant to B: its curl won't change, but you can change the direction of B arbitrarily.
Ok, i see! Many thanks for the help :D