Context :

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 , H _ { \textrm { non } } , and a magnetically , H _ { \textrm { mag } } , heated component from a non-heated component , H _ { \textrm { co } } 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 \approx 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 ( H _ { \textrm { mag } } \approx 2 pm , H _ { \textrm { non } } \approx 3 pm ) .
Considering the effect of straylight , the contribution from an atmosphere with no temperature rise to the H-index ( H _ { \textrm { co } } \approx 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 .