We can use Gauss’ law to replace p = Ɛ_{0} Δ • E in the continuity equation to get

**Δ. J + [ϐ (Ɛ _{0} Δ • E) / ϐt] = 0**

So, **Δ . (J + J _{D}) = 0**

**Where, J _{D} = Ɛ_{0} (ϐE/ϐt)**

is the displacement current density, and has the same dimensions as the current density. Since J_{D} is caused by a changing electric field, it will be non-zero between the capacitor plates.

Using Eq. [**Δ . (J + J _{D}) = 0]**, Ampere’s Law can then be modified to make Δ • (Δ x B) = 0, i.e.

**Δ x B = μ _{0} J + μ_{0} Ɛ_{0} (ϐE/ϐt)**

which is Ampere’s law as modified by Maxwell. From this we may conclude that currents and displacement currents are on an equal footing in electrodynamics, and that a changing electric field induces a magnetic field.

Returning to the problem of the charging capacitor, and assuming the electric field is uniform between the plates and zero elsewhere, Gauss’ law gives

**E = (σ / Ɛ _{0}) = (1/ Ɛ_{0}) (Q/A)**

where a is the surface charge density, Q is the charge and A the plate area. Thus between the plates the displacement current is

Hence, the total displacement current I_{D} between the plates is identical to the current I in the wires charging the plates.