The properties of very powerful extended (FR II) radio sources can be used to probe their gaseous environments. In particular, the velocity of lobe propagation and the lobe minimum energy magnetic field may be used to estimate the density of the ambient gas around a given source. A sample of 14 radio galaxies and eight radio-loud quasars with redshifts from 0 to 2 are studied in detail. The radio data are used to estimate the lobe magnetic field and the lobe propagation velocity. These are combined to obtain an estimate of the ambient gas density in the vicinity of the radio lobe using the equation of ram pressure confinement; generally, two densities are obtained for each source, one for each radio lobe. Several possible selection effects are studied in detail, including the power-redshift selection effect that arises from the fact that the sample is flux limited, and the correlation of radio spectral index with redshift. The key result is that the sources are in gaseous environments similar to those found in low-redshift clusters of galaxies. One of the sources in this study is Cygnus A, and the gaseous environments of the sources studied seem to be similar to that in the vicinity of this low-redshift, very powerful extended radio source. Not only are the typical densities obtained similar to those found in low-redshift clusters, but the composite density profile is as well. Thus, it appears that these sources lie in cluster-like gaseous environments, though the sources are observed out to relatively large redshift, having redshifts between 0 and 2. There is some evidence that the core density of the gaseous environments about the sources evolves with redshift in the sense that higher redshift systems have lower core gas densities, but the data are consistent with a constant core gas mass model in which the core density decreases and the core radius increases with redshift in such a way that the total core gas mass remains roughly constant. It does not seem likely that this result is related to the radio power-redshift selection effect, but it could be related to the radio spectral index-redshift selection effect, and a study of this and other selection effects is continuing. When a simple correction is applied to account for the radio spectral index-redshift selection effect, the negative evolution of the core gas density with redshift is only significant at about the 2 σ level. Several independent observations now indicate that powerful extended radio sources have magnetic field strengths that are lower than estimated minimum energy field strengths. Three independent measures of the offset of the magnetic field from that estimated assuming minimum energy conditions are discussed here and in a companion paper. Consistency between independent measures of the same quantity is obtained if the true magnetic field strength has an offset of about 0.25 from the minimum energy field, a result consistent with that obtained by other groups. In addition, the data presented here can be used to place a limit on the source-to-source dispersion of the offset. It is shown that most sources probably have a very similar offset; the source-to-source dispersion in the offset is likely to be less than about 15%.
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