A change in
electromotive force (emf) will be opposed by the
inertia of the charge carriers since, like all objects with mass, they prefer to be traveling at constant velocity; therefore it takes a finite time to accelerate the particle. This is similar to how a change in emf is opposed by the finite rate of change of magnetic flux in an inductor. The resulting phase lag in voltage is identical for both energy storage mechanisms, making them indistinguishable in a normal circuit. Kinetic inductance (L_{K}) arises naturally in the
Drude model of
electrical conduction considering not only the DC conductivity but also the finite relaxation time (collision time) \tau of the mobile charge carriers when it is not tiny compared to the wave period 1/f. This model defines a
complex conductance at radian frequency ω=2πf given by {\sigma(\omega) = \sigma_{1} - i\sigma_{2}}. The imaginary part, -σ2, represents the kinetic inductance. The Drude complex conductivity can be expanded into its real and imaginary components: \sigma = \frac{ne^2\tau}{m(1+i\omega\tau)} = \frac{n e^2 \tau}{m} \left(\frac{1}{1+\omega^2\tau^2}-i\frac{\omega\tau}{1+\omega^2\tau^2} \right) where m is the mass of the charge carrier (i.e. the effective
electron mass in metallic
conductors) and n is the carrier number density. In normal metals the collision time is typically \approx 10^{-14} s, so for frequencies {\omega \tau} is very small and can be ignored; then this equation reduces to the DC conductance \sigma_0 = ne^2\tau/m. Kinetic inductance is therefore only significant at optical frequencies, and in superconductors whose {\tau \rightarrow \infty}. For a superconducting wire of cross-sectional area A, the kinetic inductance of a segment of length l can be calculated by equating the total kinetic energy of the
Cooper pairs in that region with an equivalent inductive energy due to the wire's current I: \frac{1}{2}(2m_e v^2)(n_{s}lA)=\frac{1}{2}L_KI^2 where m_e is the electron mass (2m_e is the mass of a Cooper pair), v is the average Cooper pair velocity, n_{s} is the density of Cooper pairs, l is the length of the wire, A is the wire cross-sectional area, and I is the current. Using the fact that the current I = 2evn_{s}A, where e is the electron charge, this yields: L_K=\left(\frac{m_e}{2n_{s}e^2}\right)\left(\frac{l}{A}\right) The same procedure can be used to calculate the kinetic inductance of a normal (i.e. non-superconducting) wire, except with 2m_e replaced by m_e, 2e replaced by e, and n_{s} replaced by the normal carrier density n. This yields: L_K=\left(\frac{m_e}{ne^2}\right)\left(\frac{l}{A}\right) The kinetic inductance increases as the carrier density decreases. Physically, this is because a smaller number of carriers must have a proportionally greater velocity than a larger number of carriers in order to produce the same current, whereas their energy increases according to the
square of velocity. The
resistivity also increases as the carrier density n decreases, thereby maintaining a constant ratio (and thus phase angle) between the (kinetic) inductive and resistive components of a wire's
impedance for a given frequency. That ratio, \omega \tau, is tiny in normal metals up to
terahertz frequencies. In two-dimensional conductors with quadratic electron energy dispersion with an effective electron mass of m_e, the kinetic inductance is given by L_K=\left(\frac{m_e}{ne^2}\right)\left(\frac{l}{W}\right) where n is the electron density per unit area and W is the width of the two-dimensional conductor. The kinetic inductance is especially pronounced in two-dimensional conductors. In graphene where individual electron mass is zero, still the kinetic inductance exists due to the collective inertia of electrons, where it is given by L_K=\left(\frac{\pi\hbar^2}{e^2 \epsilon_F}\right)\left(\frac{l}{W}\right) Here \epsilon_F is the Fermi energy. == Applications ==