  Electric force microscopy was used to record the light-dependent impedance
spectrum and the probe transient photoconductivity of a film of butylammonium
lead iodide, BA$_{2}$PbI$_{4}$, a 2D Ruddlesden--Popper perovskite
semiconductor. The impedance spectrum of BA$_{2}$PbI$_{4}$ showed modest
changes as the illumination intensity was varied up to 1400 mW/cm$^{2}$, in
contrast with the comparatively dramatic changes seen for 3D lead-halide
perovskites under similar conditions. BA$_{2}$PbI$_{4}$'s light-induced
conductivity had a rise time and decay time of $\sim$ 100 $\mu$s, 10$^{4}$
slower than expected from direct electron-hole recombination and yet 10$^{5}$
faster than the conductivity-recovery times recently observed in 3D lead-halide
perovskites and attributed to the relaxation of photogenerated vacancies. What
sample properties are probed by electric force microscope measurements remains
an open question. A Lagrangian-mechanics treatment of the electric force
microscope experiment was recently introduced by Dwyer, Harrell, and Marohn
which enabled the calculation of steady-state electric force microscope signals
in terms of a complex sample impedance. Here this impedance treatment of the
tip-sample interaction is extended, through the introduction of a
time-dependent transfer function, to include time-resolved electrical scanned
probe measurements. It is shown that the signal in a phase-kick electric force
microscope experiment, and therefore also the signal in a time-resolved
electrostatic force microscope experiment, can be written explicitly in terms
of the sample's time-dependent resistance (i.e., conductivity).
