Derivation of the Differential Equations of the Magnetic Field and Ampère's Law

Introduction

The study of electromagnetic fields is foundational to modern physics, with Ampère’s Law serving as a cornerstone in understanding the relationship between electric currents and magnetic fields. The original Ampère’s Law, formulated in the 19th century, was a phenomenological description of the magnetic field generated by steady currents. However, this law lacked a complete theoretical framework, particularly in accounting for the effects of time-varying electric fields. The development of Maxwell’s equations in the 1860s addressed this limitation by introducing the concept of displacement current, thereby providing a comprehensive description of electromagnetic phenomena. This article focuses on the derivation of the differential equations governing the magnetic field, with particular emphasis on the role of Ampère’s Law and the inclusion of the displacement current term.

Core Content

The Integral Form of Ampère’s Law

Ampère’s Law, in its original form, relates the magnetic field $ \mathbf{B} $ to the current density $ \mathbf{J} $ through the equation:
$$ \oint_{\partial V} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} $$
Here, $ \mathbf{B} $ is the magnetic field, $ \partial V $ is the boundary of a closed surface $ V $, and $ I_{\text{enc}} $ is the total current enclosed by the surface. This equation describes the relationship between the circulation of the magnetic field around a closed loop and the enclosed current. However, this form is an integral equation, which is not immediately useful for analyzing fields in continuous media.

Transition to the Differential Form

To derive the differential form of Ampère’s Law, we apply Stokes’ Theorem, which relates the circulation of a vector field to the integral of its curl over a surface. For the magnetic field $ \mathbf{B} $, Stokes’ Theorem transforms the integral equation into:
$$ \iint_{\partial V} (\nabla \times \mathbf{B}) \cdot d\mathbf{S} = \mu_0 I_{\text{enc}} $$
This equation expresses the relationship between the curl of the magnetic field and the enclosed current. However, the integral form alone does not account for the effects of time-varying electric fields, which are critical in electromagnetic theory.

Incorporating the Displacement Current

Maxwell’s contribution to electromagnetism addressed this limitation by introducing the displacement current term to Ampère’s Law. The differential form of Ampère’s Law, which includes the displacement current, is derived as:
$$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$
Here, $ \mathbf{J} $ is the current density, $ \mathbf{E} $ is the electric field, and $ \varepsilon_0 $ is the vacuum permittivity. This equation accounts for both the conduction current $ \mathbf{J} $ and the displacement current $ \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} $, which arises from the changing electric field.

Derivation of the Differential Equation

The derivation begins by considering the integral form of Ampère’s Law and applying Stokes’ Theorem. The surface integral $ \iint (\nabla \times \mathbf{B}) \cdot d\mathbf{S} $ is then converted into a volume integral using the divergence theorem (Gauss’s Theorem), which relates the flux of a vector field through a surface to the divergence of the