  Trapped ions are among the leading candidates for quantum computing
technologies. Interfacing ion qubits in separate traps and interfacing ion
qubits with superconducting qubits are two of the many challenges to scale up
quantum computers. One approach to overcome both problems is to use a
conducting wire to mediate the Coulomb interaction between ions in different
traps, or between ions and superconducting qubits. To this end, a trapped
charged particle inducing charge on a conductor has long been modeled as a
system of equivalent lumped element electronic components. Careful
consideration reveals two assumptions in the derivation of this model which are
unjustified in many situations of interest. We identify these assumptions and
explain their implications. In addition, we introduce an improved way to use
linear relationships to describe the interaction of trapped ions with nearby
conductors. The new method is based on realistic assumptions and reproduces
results from other works that are not based on the circuit element model. It is
targeted for trouble-shooting experimental designs and allows experiments to
test and compare the accuracy of different theoretical models.
