  Hadron masses are subject to few MeV corrections arising from QED
interactions, almost entirely arising from the electric charge of the valence
quarks. The QED effects include both self-energy contributions and interactions
between the valence quarks/anti-quarks. By combining results from different
signs of the valence quark electric charge we are able to isolate the
interaction term which is dominated by the Coulomb piece, $\langle
\alpha_{\mathrm{QED}}e_{q_1}e_{\overline{q}_2}/r \rangle$, in the
nonrelativistic limit. We study this for $D_s$, $\eta_c$ and $J/\psi$ mesons,
working in lattice QCD plus quenched QED. We use gluon field configurations
that include up, down, strange and charm quarks in the sea at multiple values
of the lattice spacing. Our results, including also values for mesons with
quarks heavier than charm, can be used to improve phenomenological models for
the QED contributions. The QED interaction term carries information about meson
structure; we derive effective sizes $\langle 1/r_{\mathrm{eff}} \rangle^{-1}$
for $\eta_c$, $J/\psi$ and $D_s$ of 0.206(8) fm, 0.321(14) fm and 0.307(31) fm
respectively.
