Complex conductivity ($\hat{\mathbf{J}}(\omega )=\sigma (\omega )\hat{\mathbf{E}}(\omega )$)

Complex conductivity relates the complex amplitudes of current density and electric field at an angular frequency $\omega$ in sinusoidal response. Its modulus gives the amplitude ratio, while its argument encodes their relative phase. Both the exponential time convention and whether amplitudes are peak or root mean square values must be stated when interpreting phase or average power.