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.^[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 $p={ϵ}_0{E}^2/2$. Its force magnitude isThe 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 $C={ϵ}_{0}A/g$ 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.^[ref]