Curie–von Schweidler law

According to this law, the current decays according to a power law:

${\displaystyle I\left(t\right)\propto t^{-n},}$

where ${\displaystyle I\left(t\right)}$ is the current at a given charging time, ${\displaystyle t}$, and ${\displaystyle n}$ is the decay constant such that ${\displaystyle 0<n<1}$. Given that the dielectric has a finite conductance, the equation for current measured through a dielectric under a DC electrical field is:

${\displaystyle I\left(t\right)=at^{b}+c,}$

where ${\displaystyle a}$ is a constant of proportionality, ${\displaystyle b}$ is the decay constant (i.e., ${\displaystyle b=-n}$), and ${\displaystyle c}$ is the intrinsic conductance of the dielectric. This stands in contrast to the Debye formulation, which states that the current is proportional an exponential function with a time constant, ${\displaystyle \tau }$, according to:

${\displaystyle I\left(t\right)\propto \exp \left\{-t/\tau \right\}}$.

The Curie–von Schweidler behavior has been observed in many instances such as those shown by Andrzej Ka Johnscher[4] and Jameson et al.[5] It has been interpreted as a many-body problem by Jonscher, but can also be formulated as an infinite number of resistor-capacitor circuits. This comes from the fact that the power law can be expressed as:

${\displaystyle t^{-n}={\frac {1}{\Gamma \left(n\right)}}\int _{0}^{\infty }\tau ^{-\left(n+1\right)}e^{-t/\tau }d\tau ,}$

where ${\displaystyle \Gamma \left(n\right)}$ is the Gamma function. Effectively, this relationship shows the power law expression to be equivalent to an infinite weighted sum of Debye responses.