The distribution of electric field *E* within the general electrode system follows from the equation

$$\begin{array}{}{\displaystyle \text{div}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\epsilon \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{grad}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phi =0\phantom{\rule{thinmathspace}{0ex}},}\end{array}$$(1)

where *ε* denotes the permittivity, *φ* is scalar electric potential (*E* = −grad*φ*). Externally generated volume charge density is neglected (attracted material is not previously polarised).

To get results from numerical solution of (1) there is no need to solve the whole surface where the attraction force *F*_{a} occurs. The electric field *E* can be calculated only in one segment that consists of two electrodes with opposite charges. The reason is a repetitive pattern of electrodes [6].

The electrostatic force *F*_{e} exerted on the segment can be calculated from the distribution of *E* using the formula

$$\begin{array}{}{\displaystyle {\mathit{F}}_{\mathrm{e}}=\underset{S}{\oint}\mathit{T}\text{d}\mathit{S}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\mathit{T}=\frac{1}{2}(\mathit{E}\cdot \mathit{D})\mathit{I}+\mathit{E}\otimes \mathit{D}\phantom{\rule{thinmathspace}{0ex}},}\end{array}$$(2)

where *T* is the Maxwell stress tensor (*S* being the outward normal to complete boundary of the attracted object). Symbol *D* represents the dielectric flux density (*D* = *ε**E*), *I* is unit diagonal matrix and symbol ⊗ denotes the dyadic product [6].

Numerical calculation of (2) can be simplified assuming interdigital electrodes parallel with the surface of attracted object. The Maxwell stress tensor can be then represented as follows

$$\begin{array}{}{\displaystyle \mathit{T}=\left[\begin{array}{cc}\frac{\epsilon}{2}\left({E}_{x}^{2}-{E}_{y}^{2}\right)& \epsilon {E}_{x}{E}_{y})\\ (\epsilon {E}_{y}{E}_{x})& \frac{\epsilon}{2}\left({E}_{y}^{2}-{E}_{x}^{2}\right)\end{array}\right]\phantom{\rule{thinmathspace}{0ex}},}\end{array}$$

where *E*_{x} and *E*_{y} are electric field components. Normal direction of electrostatic force *F*_{ey} acting on the segment with electrodes of length *l* is then given

$$\begin{array}{}{\displaystyle {F}_{\mathrm{e}y}=\underset{S}{\oint}{T}_{y}\text{d}\mathit{S}=\frac{1}{2}\epsilon l\underset{0}{\overset{w+g}{\int}}({E}_{y}^{2}-{E}_{x}^{2})\text{d}x\phantom{\rule{thinmathspace}{0ex}}.}\end{array}$$(3)

Figure 6 Definition area of electroadhesion foil (values of model parameters correspond with fabricated and measured foil)

Let us also mention often used [3] but also rough calculation approach that yields the electrostatic force *F*_{ey} between the two parallel plates of a capacitor

$$\begin{array}{}{\displaystyle {F}_{\mathrm{e}y}=-\frac{\mathrm{\partial}{W}_{\mathrm{e}}}{\mathrm{\partial}y}=-\frac{1}{2}\frac{\mathrm{\partial}{E}_{y}{D}_{y}}{\mathrm{\partial}y}\underset{V}{\int}\text{d}V=-\frac{\epsilon}{2}{\left(\frac{U}{t}\right)}^{2}wl\phantom{\rule{thinmathspace}{0ex}},}\end{array}$$(4)

where *W*_{e} is total energy, *D*_{y} is electric displacement field component, *V* is volume of capacitor, *U* is applied voltage and *t* is distance between electrodes.

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