State Ampere’s Circuital law. Derive Maxwell’s fourth equation in differential form.

Ampere's circuital law: The line integral of the magnetic field intensity \(\vec H\) around any closed path equals the net current enclosed by that path:

\[ \oint\vec H\cdot d\vec l=I_{enc}\quad\Big(\text{or }\oint\vec B\cdot d\vec l=\mu_0I_{enc}\Big) \]

Derivation of Maxwell's fourth equation:

1. Write the enclosed current using the current density: \(\oint\vec H\cdot d\vec l=\int_S\vec J\cdot d\vec S\).
2. Stokes' theorem: \(\oint\vec H\cdot d\vec l=\int_S(\nabla\times\vec H)\cdot d\vec S\). So \(\nabla\times\vec H=\vec J\) ... (1)
3. Take the divergence of (1): \(\nabla\cdot\vec J=\nabla\cdot(\nabla\times\vec H)=0\). But the continuity equation requires \(\nabla\cdot\vec J=-\dfrac{\partial\rho}{\partial t}\neq0\) for time-varying fields. So (1) is incomplete.
4. Maxwell added a term \(\vec J_d\): \(\nabla\times\vec H=\vec J+\vec J_d\). Taking the divergence: \(0=\nabla\cdot\vec J+\nabla\cdot\vec J_d\), so \(\nabla\cdot\vec J_d=\dfrac{\partial\rho}{\partial t}\).
5. Using Gauss's law \(\rho=\nabla\cdot\vec D\): \(\nabla\cdot\vec J_d=\nabla\cdot\dfrac{\partial\vec D}{\partial t}\Rightarrow\vec J_d=\dfrac{\partial\vec D}{\partial t}\) (displacement current density).

\[ \boxed{\nabla\times\vec H=\vec J+\frac{\partial\vec D}{\partial t}} \]

This is Maxwell's fourth equation (the Ampere–Maxwell law). In free space: \(\nabla\times\vec B=\mu_0\varepsilon_0\dfrac{\partial\vec E}{\partial t}\). A changing electric field produces a magnetic field.