Einstein's field equations for a perfect fluid coupled to a frozen-in magnetic field are studied in the highdensity limit of gravitational collapse. The assumption of infinite electrical conductivity is used to integrate Maxwell's equations and the fluid entropy conservation equation; and the integrals obtained show that there are certain general, physically reasonable conditions under which the electromagnetic energy density can become much larger than the fluid energy density as the collapse proceeds, even when the electromagnetic field was initially very weak. The widest possible range of cases is discussed under the assumption that the equation of state is asymptotically linear. Ways in which the hypotheses used might go wrong are mentioned.