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# Coulomb force

or electric force

Force between two electric charges. The magnitude of the force math.F is proportional to the product of the two charges, math.q1 and math.q2, divided by the square of the distance math.r between them, or math.F = math.kmath.q1math.q2/math.r2, where math.k is a constant that depends on the measurement system being used. The Coulomb force can be one of repulsion, such as the force between two objects having like charges, or it can be attractive, such as the force between two objects having opposite charges.

Electrostatic force microscopy (EFM) is a type of dynamic non-contact atomic force microscopy where the electrostatic force is probed. ("Dynamic" here means that the cantilever is oscillating and does not make contact with the sample). This force arises due to the attraction or repulsion of separated charges. It is a long-ranged force and can be detected 100nm from the sample. For example, consider a conductive cantilever tip and sample which are separated a distance $z$ usually by a vacuum. A bias voltage between tip and sample is applied by an external battery forming a capacitor, C, between the two. The capacitance of the system depends on the geometry of the tip and sample. The total energy stored in that capacitor is $U = - frac\left\{1\right\}\left\{2\right\} C Delta V^2$. The battery works to maintain a constant voltage, $Delta V$, between the capacitor plates (tip and sample). By definition, taking the negative gradient of the energy gives the force. The z component of the force (the force along the axis connecting the tip and sample) is thus:

$F_\left\{electrostatic\right\} = frac\left\{1\right\}\left\{2\right\} frac\left\{partial C\right\}\left\{partial z\right\} Delta V^2$.

The electrostatic force can be probed by changing the voltage, and that force is parabolic with respect to the voltage. One note to make is that $Delta V$ is not simply the voltage difference between the tip and sample. Since the tip and sample are often not the same material, and furthermore can be subject to trapped charges, debris, etc., there is a difference between the work functions of the two. This difference, when expressed in terms of a voltage, is called the contact potential difference, $V_\left\{CPD\right\}$ This causes the apex of the parabola to rest at $Delta V = V_\left\{tip\right\} - V_\left\{sample\right\} - V_\left\{CPD\right\} = 0$. Typically, the value of $V_\left\{CPD\right\}$ is on the order of a few hundred millivolts. Forces as small as piconewtons can routinely be detected with this method.

With an electrostatic force microscope, like the atomic force microscope it is based on, the sample can be immersed in liquid.

## References

L. Kantorovich, A. Livshits, and M. Stoneham, J. Phys.:Condens. Matter 12, 795 (2000)

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