Magnetic fields of unknown origin appear to permeate the Universe on all large scales. We suggest that the winds and cosmic rays, which are produced by the spherical ejecta and jets from supernova explosions, magnetize the interstellar medium in galaxies, the intracluster medium in galaxy clusters, and the intergalactic space. We show that, if the cosmic rays generate magnetic fields in rough energy equipartition with the cosmic rays, the predicted magnetic-field strengths coincide with the ones observed. DOI: 10.1103/PhysRevD.72.123002 PACS numbers: 98.38.Am, 98.35.Nq, 98.58.Ay, 98.62.En The average magnetic field (MF) of the interstellar medium (ISM) of our Galaxy (B MW 5 G) corresponds to an energy density of 0:5 eV cm ÿ3 , in good agreement with the energy density of galactic cosmic rays (CRs) [1]. This provides a strong hint of a common origin and of energy equipartition [2], though other theories of the origin of galactic fields-such as dynamo amplification of primordial seed fields-have been proposed [3].The origin of intergalactic MFs, both within and without galaxy clusters, is also undecided. Radio observations of galaxy clusters indicate that the intra-cluster medium (ICM) between the galaxies is permeated by intense MFs [4]. Nearby clusters are seen to have a ''radio halo'' with a distribution similar to that of the cluster gas, observed in x rays. These halos are produced by synchrotron emission from CR electrons spiraling in the cluster's MF, while the x rays are electron bremsstrahlung. Measurements of the Faraday rotation of linearly polarized radio emission traversing the cluster's medium, in combination with x-ray data, support the existence of cluster MFs of a few G [5,6]. The mapping of the Faraday rotation reveals that the clusters' MFs are turbulent with a Kolmogoroff power spectrum on a variety of scales [7].The MF between clusters and isolated galaxies in the inter-galactic medium (IGM) is not known. Speculations on its value range from nearly a G to a pG. Low-level radio emission was detected from the IGM around Coma [6] and from the IGM in large-scale filaments of galaxies [8]. The estimated field strengths are of the order of several hundred nG.Theories of the origin of MFs in the ICM and IGM include cosmic shocks [9], ionization fronts [10] and outflows from primeval galaxies [4], quasars and/or radio galaxies [11]. In particular, Kronberg et al. have estimated [12] that ''giant'' extragalactic radio sources, powered by accretion onto massive black holes (M > 10 8 M ), inject E B 10 60-61 erg of magnetic energy into radio lobes, and have argued that the expansion and diffusion of these Mpcscale lobes could have magnetized a large fraction of the IGM. Assuming that in the accretion 1% of M is released in the form of magnetic energy, they estimated a mean B IGM 40 nG at redshift z 2. This value later evolves as 1 z 2 due to cosmic expansion, yielding a two-ordersof-magnitude smaller IGM energy density, and a one-order-of-magnitude smaller B IGM at z 0.Concerning the ICM, it was su...