# Permittivity ε

For a dielectric material, the areal charge density (Coulomb/m^{2}) that is generated by an electric field (volt/m).

\begin{align} \label{eq:10900a} \textsf{Units} &= \frac{\textsf{Coulomb/m}^2}{\textsf{volt/m}} \nonumber \\[0.7em]%eqn_interline_spacing &= \frac{\textsf{Coulomb/volt}}{\textsf{m}} \nonumber \\[0.7em]%eqn_interline_spacing &= \frac{\textsf{Farad}}{\textsf{m}} \nonumber \end{align}

The permittivity of free space (\( \varepsilon_o \) = 8.85419 × 10^{−12} F/m) is used as a reference.

For a piezoelectric material, the permittivity (and dielectric constant) depends on the state of strain. Two distinct permittivities are possible — the permittivity when the piezoelectric is completely unconstrained (i.e., the *free* permittivity \( \varepsilon^T \) and the permittivity when the piezoelectric is completely constrained (i.e., the *clamped (blocked)* or *constant-strain* permittivity \( \varepsilon^S \)). These two permittivities are related by the electromechanical coupling coefficient к (Waanders, equation A15, p. 84):

\begin{align} \label{eq:10901a} \varepsilon^S = \varepsilon^T \left( 1 - \kappa^2 \right) \end{align}

Since \( \kappa \) is always between 0 and 1, \( \varepsilon^S \) ≤ \( \varepsilon^T \). In a transducer the piezoelectric ceramic is neither truly clamped nor truly free. Therefore, the permittivity will have an intermediate value between \( \varepsilon^S \) and \( \varepsilon^T \).

Similarly, the dielectric constant will have a value between unconstrained \( K^S \) and blocked \( K^T \). "So far as one can speak of the "true" dielectric constant of the piezoelectric crystal, the value at constant strain [\( K^S \)] is the proper one to use." (Cady (1), p. 161)

Notes:

- The premittivity may depend on the electric field strength, the exciting frequency, the temperature, and other factors.
- A state of constant strain cannot be practically achieved by physically restraining the piezoelectric ceramic. However, it can be achieved by exciting at a very high frequency where the inertial effects become so large that the ceramic effectively cannot vibrate (i.e., there are no strains). (See Cady (1), pp. 328, 572.) The same situation occurs when a single degreeoffreedom spring-mass system is excited at a frequency that is very high compared to its resonant frequency — the mass will not move from its rest position.

See dielectric constant.