Section: 6 | Permittivity (Dielectric Constant) of Water at Various Frequencies |
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The permittivity of liquid water in the radio frequency and microwave regions can be represented by the Debye equation (Refs. 1 and 2):

$\begin{array}{c}{\epsilon}^{\prime}={\epsilon}_{\infty}+\frac{{\epsilon}_{s}-{\epsilon}_{\infty}}{1+{\omega}^{2}{\tau}^{2}}\\ {{\epsilon}^{\prime}}^{\prime}=\frac{({\epsilon}_{s}-{\epsilon}_{\infty})\omega \tau}{1+{\omega}^{2}{\tau}^{2}}\end{array}$

where *ε* = *ε*´ + i *ε*˝ is the (complex) relative permittivity (i.e., the absolute permittivity divided by the permittivity of free space *ε*_{0} = 8.854·10^{–12} F m^{–1}). Here *ε*_{s} is the static permittivity (see Ref. 3 and the table “Thermophysical Properties of Water and Steam” in this section); *ε*_{∞} is a parameter describing the permittivity in the high-frequency limit; *τ *is the relaxation time for molecular orientation; and *ω* = 2π*f* is the angular frequency. The values in Table 2 have been calculated from parameters given in Ref. 2, as shown in Table 1. Column definitions for Table 2 are as follows.

Column heading |
Definition |

Frequency |
Frequency, in kHz, MHz, or GHz |

ε´ (temperature) |
Real part of the relative permittiviy at the indicated temperature in °C |

ε˝ (temperature) |
Imaginary part of the relative permittivity at the indicated temperature in °C |

Other useful quantities that can be calculated from the values in the table are the loss tangent:

tan *δ* = *ε*"/*ε*'

and the absorption coefficient *α* which describes the power attenuation per unit length (*P = P*_{0}e^{–αl}):

$\alpha =\frac{\pi \text{}f{\epsilon}^{\u2033}}{c\sqrt{{\epsilon}^{\prime}\text{}}}$

and *c* is the speed of light. The last equation is valid when *ε*˝/*ε*´<<1.

- Fernández, D. P., Mulev, Y., Goodwin, A. R. H., and Levelt Sengers, J. M. H
*., J. Phys. Chem. Ref. Data*24, 33, 1995. [https://doi.org/10.1063/1.555977] - Kaatze, U.
*, J. Chem. Eng. Data*34, 371, 1989. [https://doi.org/10.1021/je00058a001] - Archer, D. G., and Wang, P.,
*J. Phys. Chem. Ref. Data*19, 371, 1990. [https://doi.org/10.1063/1.555853]

0 °C | 25 °C | 50 °C | |

ε_{∞} | 5.7 | 5.2 | 4.0 |

τ/ps | 17.67 | 8.27 | 4.75 |

Frequency | ε´ (0 °C) | ε˝ (0 °C) | ε´ (25 °C) | ε˝ (25 °C) | ε´ (50 °C) | ε˝ (50 °C) | |

Water | 0 | 87.90 | 0.00 | 78.36 | 0.00 | 69.88 | 0.00 |

Water | 1 kHz | 87.90 | 0.00 | 78.36 | 0.00 | 69.88 | 0.00 |

Water | 1 MHz | 87.90 | 0.01 | 78.36 | 0.00 | 69.88 | 0.00 |

Water | 10 MHz | 87.90 | 0.09 | 78.36 | 0.04 | 69.88 | 0.02 |

Water | 100 MHz | 87.89 | 0.91 | 78.36 | 0.38 | 69.88 | 0.20 |

Water | 200 MHz | 87.86 | 1.82 | 78.35 | 0.76 | 69.88 | 0.39 |

Water | 500 MHz | 87.65 | 4.55 | 78.31 | 1.90 | 69.87 | 0.98 |

Water | 1 GHz | 86.90 | 9.01 | 78.16 | 3.79 | 69.82 | 1.96 |

Water | 2 GHz | 84.04 | 17.39 | 77.58 | 7.52 | 69.65 | 3.92 |

Water | 3 GHz | 79.69 | 24.64 | 76.62 | 11.13 | 69.36 | 5.85 |

Water | 4 GHz | 74.36 | 30.49 | 75.33 | 14.58 | 68.95 | 7.75 |

Water | 5 GHz | 68.54 | 34.88 | 73.73 | 17.81 | 68.45 | 9.62 |

Water | 10 GHz | 42.52 | 40.88 | 62.81 | 29.93 | 64.49 | 18.05 |

Water | 20 GHz | 19.56 | 30.78 | 40.37 | 36.55 | 52.57 | 28.99 |

Water | 30 GHz | 12.50 | 22.64 | 26.53 | 33.25 | 40.57 | 32.74 |

Water | 40 GHz | 9.67 | 17.62 | 18.95 | 28.58 | 31.17 | 32.43 |

Water | 50 GHz | 8.28 | 14.34 | 14.64 | 24.53 | 24.42 | 30.47 |