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In the Beginning, the Biermann Battery In class, in deriving Ohm’s Law (the “difference” equation between the ion and electron momentum equations), we dropped a term that looked like @ @xs (-enehwerwesi) (5) Restore this term, but keep only the diagonal components of the electron stress tensor, i.e., set nemehwerwesi = Pers, where me is the mass of the electron, Pe is the electron partial pressure, and rs is the Kronecker delta. (In other words, ignore the viscous contribution to the stress tensor and keep only the pressure contribution). 2 Derive the slightly more general form of the induction equation @~B @t = r × (~v × ~B ) + c2 4 r2B + c n2 ee rPe × rne (6) The last term on the right-hand-side is the term introduced by Ludwig Biermann to generate magnetic fields from material that is initially field-free. It relies upon the contours of constant electron pressure being misaligned with the contours of constant electron density to pressure- accelerate electrons away from ions in such a way as to produce current (which in turn produces field). The action is similar to that of a battery (which relies on chemical reactions to accelerate electrons away from ions), and for this reason we call this term the Biermann battery term. It has been invoked to generate truly primordial magnetic field in proto-galactic plasma (see Kulsrud’s (1997) article in “Critical Dialogues in Cosmology.”)
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