Aims: We investigate the relationship between the photospheric magnetic field and the emission of the mid chromosphere of the Sun. Methods: We simultaneously observed the Stokes parameters of the photospheric iron line pair at 630.2 nm and the intensity profile of the chromospheric Ca II H line at 396.8 nm in a quiet Sun region at a heliocentric angle of 53°. Various line parameters have been deduced from the Ca II H line profile. The photospheric magnetic field vector has been reconstructed from an inversion of the measured Stokes profiles. After alignment of the Ca and Fe maps, a common mask has been created to define network and inter-network regions. We perform a statistical analysis of network and inter-network properties. The H-index is the integrated emission in a 0.1 nm band around the Ca core. We separate a non-magnetically, Hnon, and a magnetically, Hmag, heated component from a non-heated component, Hco in the H-index. Results: The average network and inter-network H-indices are equal to 12 and 10 pm, respectively. The emission in the network is correlated with the magnetic flux density, approaching a value of H ≈ 10 pm for vanishing flux. The inter-network magnetic field is dominated by weak field strengths with values down to 200 G and has a mean absolute flux density of about 11 Mx cm-2. Conclusions: We find that a dominant fraction of the calcium emission caused by the heated atmosphere in the magnetic network has non-magnetic origin (Hmag≈2 pm, Hnon≈3 pm). Considering the effect of straylight, the contribution from an atmosphere with no temperature rise to the H-index (Hco≈6 pm) is about half of the observed H-index in the inter-network. The H-index in the inter-network is not correlated to any property of the photospheric magnetic field, suggesting that magnetic flux concentrations have a negligible role in the chromospheric heating in this region. The height range of the thermal coupling between the photosphere and low/mid chromosphere increases in presence of magnetic field. In addition, we demonstrate that a poor signal-to-noise level in the Stokes profiles leads to a significant over-estimation of the magnetic field strength.