What is Ampere's circuital law and how is it written in differential (point) form?

In one line

Ampere's circuital law states that the line integral of magnetic field around any closed loop equals \(\mu_0\) times the net current passing through that loop; in point form, \(\nabla\times\vec{B} = \mu_0\vec{J}\) for steady currents.

Explained simply

Walk once around a current-carrying wire with a compass. The magnetic field circles the wire, and if you add up "field times distance" all the way round, the answer depends only on how much current passes through your path, not on the path's shape. Currents outside the loop add nothing.

How it works

1. Integral form: \(\oint_C\vec{B}\cdot d\vec{l} = \mu_0 I_{enc}\).
2. Write the current using current density: \(I_{enc} = \int_S\vec{J}\cdot d\vec{S}\).
3. Apply Stokes' theorem to the left side: \(\oint_C\vec{B}\cdot d\vec{l} = \int_S(\nabla\times\vec{B})\cdot d\vec{S}\).
4. Since this holds for any surface, \(\nabla\times\vec{B} = \mu_0\vec{J}\), or \(\nabla\times\vec{H} = \vec{J}\) with \(\vec{B} = \mu_0\vec{H}\).
5. Limitation: taking divergence gives \(\nabla\cdot\vec{J} = 0\), which fails when charge builds up (e.g. a charging capacitor). Maxwell fixed it by adding displacement current: \(\nabla\times\vec{H} = \vec{J} + \dfrac{\partial\vec{D}}{\partial t}\).

Key formula

\(\oint\vec{B}\cdot d\vec{l} = \mu_0 I_{enc}\); \(\nabla\times\vec{H} = \vec{J}\) (static), \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\) (general). \(\vec{B}\): magnetic flux density (T), \(\vec{H}\): magnetic field intensity (A/m), \(\mu_0 = 4\pi\times10^{-7}\) T m/A, \(I\): current (A), \(\vec{J}\): current density (A/m²).

Quick example

Long straight wire, \(I = 10\) A, at \(r = 5\) cm: take a circle of radius \(r\); \(B\) is tangential and constant, so \(B\cdot2\pi r = \mu_0 I\) and \(B = \mu_0 I/2\pi r = (2\times10^{-7}\times10)/0.05 = 4\times10^{-5}\) T.

Common mistakes

• Using the total current in the circuit rather than the current crossing the chosen loop.
• Applying the divergence theorem instead of Stokes' theorem in the derivation.
• Forgetting the displacement-current term when the question is about Maxwell's equations.

How MU asks it

Asked as "State Ampere's circuital law and obtain its differential form" (5 marks), or "Explain the inconsistency in Ampere's law and how Maxwell corrected it" (5 marks). Write the statement, integral form, Stokes step, final point form, and then the Maxwell correction.