  We present a charge distribution-based emission model that calculates the
infrared spectrum of fullerenes (C$_{60}$). Analysis of the modelled spectrum
of C$_{60}$ in various charge states shows that the relative intensity of the
features in the 5-10 $\mu$m versus 15-20 $\mu$m can be used to probe the
C$_{60}$ charge state in interstellar spectra. We further used our model to
simulate emission from polycyclic aromatic hydrocarbons (PAHs) and C$_{60}$ at
five positions in the cavity of reflection nebula NGC~7023. Specifically, we
modelled the 6.2/11.2 band ratio for circumcoronene and circumcircumcoronene
and the 7.0/19.0 band ratio for C$_{60}$ as a function of the ionization
parameter $\gamma$. A comparison of the model results with the observed band
ratios shows that the $\gamma$ values in the cavity do not vary significantly,
suggesting that the emission in the cavity does not originate from locations at
the projected distances. Furthermore, we find that the C$_{60}$ derived
$\gamma$ values are lower than the PAH-derived values by an order of magnitude.
We discuss likely scenarios for this discrepancy. In one scenario, we attribute
the differences in the derived $\gamma$ values to the uncertainties in the
electron recombination rates of PAHs and C$_{60}$. In the other scenario, we
suggest that PAHs and C$_{60}$ are not co-spatial resulting in different
$\gamma$ values from their respective models. We highlight that experiments to
determine necessary rates will be required in validating either one of the
scenarios.
