The equation for electrostatic forces between a probe and a surface having different potentials is given by⁹:

It can be determined from Equation 4-5, that the change in resonant frequency is proportional to the derivative of the force. Thus, provided that the potential between the probe and surface is not zero, the change in the first derivative of the force is proportional to the changes in capacitance as a function of the second derivative of Z spacing.

Figure 4-14 illustrates the effect of an electric field on a vibrating cantilever as it is scanned across a region of electric charge. The magnitude and direction of the amplitude changes, or resonance frequency shift, depends on the polarity of charges on the sample and the polarity of charges on the probe.

FIGURE 4-14 The probe vibration amplitude at a set frequency changes as the probe moves through the electric field.

In a Magnetic Force Microscope (MFM), a probe coated with a magnetic film is vibrated slightly above a surface while scanning, illustrated in Figure 4-15.