We concentrate on astrophysical batteries here and discuss primordial magnetic fields arising from the early universe in Section 7. The basic problem any battery has to address is how to produce finite currents from zero currents? Most astrophysical mechanisms use the fact that positively and negatively charged particles in a charge-neutral universe, do not have identical properties. For example, if one considered a gas of ionized hydrogen, then the electrons have a much smaller mass compared to protons. Thus that for a given pressure gradient of the gas the electrons tend to be accelerated much more than the ions. This leads in general to an electric field, which couples back positive and negative charges. If such a thermally generated electric field has a curl, then from Faraday's law of induction a magnetic field can grow. The resulting battery effect, known as the Biermann battery, was first proposed as a mechanism for the thermal generation of stellar magnetic fields .
The thermally generated electric field is given by Ebier = - pe /e ne got by balancing the forces on the electrons, due to pressure gradient and the electric field and assuming the protons are much more massive than the electrons. The curl of this term leads to an extra source term in the induction equation, which if we adopt pe = ne kBT, where kB is the Boltzmann constant, gets modified to,
Therefore over and above the usual flux freezing and diffusion terms we have a source term which is nonzero if and only if the density and temperature gradients, ne and T, are not parallel to each other. In the cosmological context, such non-parallel density and temperature gradients can arise in a number of ways. For example, in cosmic ionization fronts produced when the first ultraviolet photon sources, like starbursting galaxies and quasars, turn on to ionize the intergalactic medium (IGM), the temperature gradient is normal to the front. However, a component to the density gradient can arise in a different direction, if the ionization front is sweeping across arbitrarily laid down density fluctuations, which will later collapse to form galaxies and clusters. Such density fluctuations, will in general have no correlation to the source of the ionizing photons, resulting in a thermally generated electric field which has a curl, and magnetic fields correlated on galactic scales can grow. After compression during galaxy formation, they turn out to have a strength B ~ 3 × 10-20G . This scenario has in fact been confirmed in detailed numerical simulations of IGM reionization . The Biermann battery has also been shown to generate both vorticity and magnetic fields in oblique cosmological shocks which arise during cosmological structure formation .
The asymmetry in the mass of the positive and negative charges can also lead to battery effects during the interaction of radiation with ionized plasma. Note that the Thomson cross section for the scattering of photons with charged particles depend inversely on the mass of the particle. So the electron component of an ionized plasma is more strongly coupled with radiation than the proton component. Suppose one has a rotating fluid element in the presence of a radiation bath. The interaction with photons will brake the velocity of the electron component faster than the proton component and set up a relative drift and hence lead to magnetic field generation . In the modern context, second order effects during recombination also leads to both vorticity and magnetic field generation due to the - e / p scattering asymmetry. The resulting magnetic fields are again very small B ~ 10-30 G on Mpc scales upto B ~ 10-21 G at parsec scales .
As can be gleaned from the few examples considered above all battery mechanisms give only very small fields on the cosmological scales much smaller than the observed fields. One therefore needs some form of dynamo action to amplify these fields further. Of course, seed fields for dynamos in larger scale objects like galaxies and clusters can arise not necessarily due to battery effects, but also due to more rapid dynamo generation in objects with a much shorter dynamical timescale, like stars and AGN, and subsequent ejection of these fields in to the interstellar and intra cluster media (cf.  for reviews). In this case much larger seed fields say for the galactic dynamo B ~ 10-9 G are possible . The down side is the as yet unresolved question as to how magnetized gas say ejected in a supernovae or AGN is mixed with unmagnetized gas in the protogalaxy, and how this mixing affects the coherence scale of the field? And even in this case one still needs dynamos to work efficiently in stars and AGN to generate the seed field, and in galaxies and clusters to further amplify and maintain it against decay.