A parallel plate capacitor is charged by connecting it to a battery through a resistor. If I is the current in the circuit, then in the gap between the plates :

When the capacitor is being charged, a conduction current I flows through the external circuit (battery → resistor → plates). Inside the dielectric gap there is no physical charge carriers, so no conduction current can exist there.

Maxwell introduced the concept of **displacement current** to preserve the continuity of current. It is defined as

$I\_d\; =\; \epsilon ₀ \backslash frac\{d\backslash Phi\_E\}\{dt\}$

where Φ_E is the electric flux through the surface bounded by the circuit loop and ε₀ is the permittivity of free space.

The electric flux through the capacitor plates is related to the electric field E and the plate area A:

$\backslash Phi\_E\; =\; E A$

Since the field between the plates is E = V/d (with d the plate separation) and the voltage V across the capacitor is changing as it charges, the rate of change of flux becomes

$\backslash frac\{d\backslash Phi\_E\}\{dt\}=A\backslash frac\{dE\}\{dt\}=A\backslash frac\{d\}\{dt\}\backslash !\backslash left(\backslash frac\{V\}\{d\}\backslash right)=\backslash frac\{A\}\{d\}\backslash frac\{dV\}\{dt\}$

The capacitance of a parallel‑plate capacitor is C = ε₀ A/d, so

$\backslash frac\{A\}\{d\}=\; \backslash frac\{C\}\{\epsilon ₀\}$

Substituting this into the expression for I_d gives

$I\_d\; =\; \epsilon ₀ \backslash frac\{C\}\{\epsilon ₀\}\backslash frac\{dV\}\{dt\}=\; C\backslash frac\{dV\}\{dt\}$

But the conduction current that charges the capacitor is

$I\; =\; \backslash frac\{dQ\}\{dt\}=\; \backslash frac\{d\}\{dt\}(C\; V)=\; C\backslash frac\{dV\}\{dt\}$

Therefore

$I\_d\; =\; I$

The direction of the displacement current follows the direction of the changing electric field, which is from the positively charged plate toward the negatively charged plate. This is exactly the same direction as the conduction current I in the external circuit.

Consequently, in the gap between the plates there is a displacement current whose magnitude equals the circuit current I and whose direction is the same as I.