= Solution
\b[Use <Kelvin probe force microscopy> to quantify a local contact-potential difference.] A conducting tip and sample form a position-dependent capacitor. Apply a <voltage> with adjustable direct and small alternating components,
$$
V(t)=V_{\rm dc}-V_{\rm CPD}+V_{\rm ac}\cos\omega t.
$$
The electrostatic <force> is proportional to $C'(z)V(t)^2/2$, so its component at $\omega$ is proportional to
$$
F_\omega=C'(z)(V_{\rm dc}-V_{\rm CPD})V_{\rm ac}\cos\omega t
$$
up to the chosen force-direction convention. A <lock-in amplifier> detects this component and a second controller adjusts $V_{\rm dc}$ to null it. \b[At the null, $V_{\rm dc}=V_{\rm CPD}$.] The compensation <voltage> becomes the potential image.<Scanning Probe Microscopy Modes by Bruker>[Surface Potential Imaging: conducting probe and electrostatic null measurement]{ref}
A lift pass can reduce direct topographic interactions. Calibration on a reference with a known <work function> is required for an absolute work-function interpretation, and the sign convention must state which electrode is biased. On a <semiconductor> the result can include surface band bending, trapped charge and surface chemistry, so it need not equal a simple map of bulk electrochemical potential.
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