• Tags: displacement-current, Ampere, capacitor
• Difficulty: Moderate

Ampere's circuital law (original): line integral of B.dl = $mu_0$$I_{enclosed}$ works for steady currents but fails for time-varying fields. Consider a charging capacitor: a wire carries current I to the plates, but between the plates there is no conduction current. Yet the magnetic field must be continuous. Maxwell's resolution: a changing electric field between the plates acts as a "displacement current" $I_d$ = $epsilon_0$ * d$\frac{Phi_E}{dt}$. Between parallel plates: E = $\frac{Q}{epsilon_0*A}$ = $\frac{sigma}{epsilon_0}$, so $Phi_E$ = EA = $\frac{Q}{epsilon_0}$. Thus $I_d$ = $epsilon_0$ * d$\frac{Q/epsilon_0}{dt}$ = $\frac{dQ}{dt}$ = $I_c$ (the conduction current in the wire). The modified Ampere-Maxwell law: line integral of B.dl = $mu_0$*($I_c$ + $I_d$). Displacement current exists wherever the electric field changes with time, even in empty space. It is not a real flow of charges but produces the same magnetic effects.