Maxwell's Equations
Table of Contents
Maxwell’s Equations represent the fundamental laws of electromagnetism.
There are two common forms of Maxwell’s Equations: the differential form and the integral form.
Differential form #
In differential form, Maxwell’s equations are given by:
$$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} $$
$$ \nabla \cdot \mathbf{B} = 0 $$
$$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$
$$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$
Where:
- \(\small \mathbf{E} \) is the electric field vector,
- \(\small \mathbf{B} \) is the magnetic field vector,
- \(\small \rho \) is the electric charge density,
- \(\small \varepsilon_0 \) is the vacuum permittivity (electric constant),
- \(\small \mu_0 \) is the vacuum permeability (magnetic constant), and
- \(\small \mathbf{J} \) is the electric current density.
Respectively, these equations are Gauss’s Law, Gauss’s Law for Magnetism, Faraday’s Law, and Ampère’s Law with Maxwell’s Addition.
In a vacuum (where there is no charge or current), these simplify to:
$$ \nabla \cdot \mathbf{E} = 0 $$ $$ \nabla \cdot \mathbf{B} = 0 $$ $$ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}$$ $$ \nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$
Integral form #
In integral form, Maxwell’s equations are given by:
$$ \oiint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = \frac{Q}{\varepsilon_0} = \frac{1}{\varepsilon_0}\iiint_{\Omega} \rho \ dV $$ $$ \oiint_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 $$ $$ \oint_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{l} = - \frac{\text{d}}{\text{d} t} \iint_{\Sigma} \mathbf{B} \cdot d\mathbf{S} $$ $$ \oint_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{l} = \mu_0 \iint_{\Sigma} \mathbf{J} \cdot d\mathbf{S} + \mu_0 \varepsilon_0 \frac{\text{d}}{\text{d} t} \iint_{\Sigma} \mathbf{E} \cdot d\mathbf{S} $$
Where:
- \( \small \mathbf{E} \) is the electric field vector,
- \(\small \mathbf{B} \) is the magnetic field vector,
- \(\small Q \) is the electric charge enclosed by the volume \(\small \Omega \),
- \( \mathbf{J} \) is the electric current density,
- \( \oiint_{\partial \Omega} \) is the surface integral over the closed surface \( \partial \Omega \),
- \( \iiint_{\Omega} \) is the volume integral over the volume \(\small \Omega \),
- \( \rho \) is the electric charge density,
- \( \oint_{\partial \Sigma} \) is the line integral over the closed curve \( \partial \Sigma \),
- \( \iint_{\Sigma} \) is the surface integral over the surface \(\small \Sigma \),
- \( \varepsilon_0 \) is the vacuum permittivity (electric constant),
- \( \mu_0 \) is the vacuum permeability (magnetic constant), and
- \( d \mathbf{l} \), \( d \mathbf{S} \), and \( dV \) are the line, surface, and volume elements, respectively.
In a vacuum (where there is no charge or current), these simplify to:
$$ \oiint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{S} = 0 $$ $$ \oiint_{\partial \Omega} \mathbf{B} \cdot d\mathbf{S} = 0 $$ $$ \oint_{\partial \Sigma} \mathbf{E} \cdot d\mathbf{l} = - \frac{\text{d}}{\text{d} t} \iint_{\Sigma} \mathbf{B} \cdot d\mathbf{S} $$ $$ \oint_{\partial \Sigma} \mathbf{B} \cdot d\mathbf{l} = \mu_0 \varepsilon_0 \frac{\text{d}}{\text{d} t} \iint_{\Sigma} \mathbf{E} \cdot d\mathbf{S} $$