The Crab Nebula emits bright non-thermal radiation from radio to the most energetic photons. The underlying physical model of a relativistic wind from the pulsar terminating in a hydrodynamic standing shock remains unchanged since the early 1970s. In this model, an increase of the toroidal magnetic field downstream from the shock is expected. We introduce a detailed radiative model to calculate non-thermal synchrotron and inverse Compton as well as thermal dust emission self-consistently to compare quantitatively with observational data. Special care is given to the radial dependence of electron and seed field density. The radiative model is used to estimate the parameters of electrons and dust in the nebula. A combined fit based upon a $\chi^2$ minimisation reproduces successfully the complete data set used. For the best-fitting model, the energy density of the magnetic field dominates over the particle energy density up to a distance of $\approx 1.3~r_s$ ($r_s$: distance of the termination shock from the pulsar). The very high energy (VHE: $E>100$ GeV) gamma-ray spectra set the strongest constraints on the radial dependence of the magnetic field: $B(r)=(264\pm9)~\mu\mathrm{G} (r/r_s)^{-0.51\pm0.03}$. The reconstructed magnetic field and its radial dependence indicates a ratio of Poynting to kinetic energy flux $\sigma\approx 0.1$ at the termination shock, $\approx 30$ times larger than estimated up to now. Consequently, the confinement of the nebula would require additional mechanisms to slow down the flow through e.g. excitation of small-scale turbulence with possible dissipation of magnetic field.