Displacement Current — Question 6

Question 6

Leakage and displacement currents are in quadrature under sinusoidal voltage

A uniform material has conductivity sigma and permittivity epsilon. Its field is $E=E_0\cos\omega t$. Use $J_c=\sigma E$ and $J_d=\epsilon\partial_tE$.

1. Find both currents and the total current-density amplitude.

2. Find the crossover angular frequency at which their amplitudes are equal.

Question 6 - Solution

Leakage and displacement currents are in quadrature under sinusoidal voltage

(a) Method and calculation

$$J_c=\sigma E_0\cos\omega t,\quad J_d=-\omega\epsilon E_0\sin\omega t.$$$$\boxed{J_{\mathrm{total,peak}}=E_0\sqrt{\sigma^2+(\omega\epsilon)^2}}.$$

(b) Interpretation and further calculation

$$J_{d,\mathrm{peak}}/J_{c,\mathrm{peak}}=\omega\epsilon/\sigma.$$$$\boxed{\omega_{\mathrm{cross}}=\sigma/\epsilon}.$$

Physical check

Amplitudes add in quadrature, not arithmetically, because one current tracks field and the other tracks its slope.