State and explain Maxwell’s fourth equation.

Statement (Ampere–Maxwell law): The line integral of the magnetic field \(\vec H\) around a closed path equals the total current, conduction plus displacement, passing through the surface bounded by that path.

\[ \text{Integral form: } \oint\vec H\cdot d\vec l=\int_S\Big(\vec J+\frac{\partial\vec D}{\partial t}\Big)\cdot d\vec S \]

\[ \text{Differential form: } \nabla\times\vec H=\vec J+\frac{\partial\vec D}{\partial t} \]

Explanation and derivation:

• Ampere's circuital law: \(\oint\vec H\cdot d\vec l=I=\int\vec J\cdot d\vec S\). By Stokes' theorem this becomes \(\nabla\times\vec H=\vec J\).
• Taking the divergence gives \(\nabla\cdot\vec J=\nabla\cdot(\nabla\times\vec H)=0\). This contradicts the continuity equation \(\nabla\cdot\vec J=-\partial\rho/\partial t\) for time-varying fields, for example in a charging capacitor.
• Maxwell added the displacement current density \(\vec J_d\): \(\nabla\times\vec H=\vec J+\vec J_d\). Then \(0=\nabla\cdot\vec J+\nabla\cdot\vec J_d\), so \(\nabla\cdot\vec J_d=\partial\rho/\partial t=\nabla\cdot\dfrac{\partial\vec D}{\partial t}\) (using \(\nabla\cdot\vec D=\rho\)). Therefore \(\vec J_d=\dfrac{\partial\vec D}{\partial t}\).

Significance: A magnetic field is produced not only by conduction current but also by a time-varying electric field (displacement current). This term, together with Faraday's law, predicts electromagnetic waves travelling at \(c=1/\sqrt{\mu_0\varepsilon_0}\). In free space (\(\vec J=0\)): \(\nabla\times\vec B=\mu_0\varepsilon_0\,\partial\vec E/\partial t\).