James Clerk Maxwell was the first to recognize that not only currents, but also time-varying electric fields, can generate magnetic fields.
Starting from Ampère’s law and replacing the current by $I = {\mathrm dQ}/{\mathrm dt}$, we obtain:
\[
\oint \vec{B}\cdot d\vec{s} = \mu_0 \frac{\mathrm dQ}{\mathrm dt}
\]
From the relationship between charge and electric field in a capacitor (see earlier equation), extended to infinitesimal surface elements $\mathrm d\vec{A}$, we get:
\[
\oint \vec{B}\cdot d\vec{s} = \mu_0 \varepsilon_0 \int \frac{\mathrm d\vec{E}}{\mathrm dt}\, \mathrm d\vec{A}
\]
Using Stokes’ theorem on the left-hand side yields:
\[
\int \mathrm{rot}\,\vec{B}\cdot d\vec{A} = \mu_0 \varepsilon_0 \int \frac{\mathrm d\vec{E}}{\mathrm dt}\, \mathrm d\vec{A}
\]
Since the infinitesimal surface elements $d\vec{A}$ cancel, we arrive at the expression for the Maxwell displacement current:
\[
\boxed{\mathrm{rot}\,\vec{B} = \mu_0 \varepsilon_0 \frac{\mathrm d\vec{E}}{\mathrm dt}}
\]
This states that a changing electric field produces a magnetic field, even without the presence of a conduction current. One can picture magnetic field lines forming around the time-varying electric field of a charging or discharging capacitor.
This is completely analogous to the induction of an electric field by a changing magnetic field. Thus, electric and magnetic fields can generate each other, which is the essential foundation for the existence of electromagnetic waves.