= Continuum electromechanics
<Continuum electromechanics> couples <electromagnetic fields> to the <motion> and deformation of materials modeled as continuous media. The mechanical state changes the fields through geometry and material properties; electric and magnetic stresses act back on the material. A consistent model therefore combines field equations, mechanical balance laws, boundary conditions and constitutive relations.<Continuum Electromechanics by James R. Melcher>[Preface and chapters 2–3, field descriptions and electromagnetic forces]{ref}
An ideal parallel-plate <capacitor> gives a simple example. Ignore edge fields and take a vacuum gap $g$, plate area $A$ and applied <electric potential difference> $V$. The <electric field> magnitude is $E=V/g$, and the <Maxwell stress tensor> gives an attractive <electrostatic pressure> $p=\epsilon_0E^2/2$. Its force magnitude is
$$
F=\frac{\epsilon_0AV^2}{2g^2}.
$$
The displacement changes the electrical force, so a suspended plate requires a coupled electrical and mechanical equilibrium calculation. A <spring> supplying the opposing force is not merely an external geometric detail.
The energy bookkeeping also depends on the constraint. At fixed charge, differentiating $Q^2/(2C)$ with the <capacitance> $C=\epsilon_0A/g$ gives the force. At fixed <electric potential difference>, the voltage source exchanges <energy> as the plate moves and must be included. Applying only the field-energy derivative while silently changing this constraint can reverse the inferred force.<Continuum Electromechanics by James R. Melcher>[Chapter 3, electromagnetic forces and thermodynamics of electromechanical coupling]{ref}
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