# Dictionary:Maxwell’s equations

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The fundamental differential equations governing the behavior of electrical and magnetic fields. The four differential equations to which electric and magnetic fields are subject at every ordinary point. In SI units,

${\displaystyle \nabla \times {\textbf {E}}=-{\frac {\partial {\textbf {B}}}{\partial t}},}$
${\displaystyle \nabla \cdot {\textbf {D}}=\rho ,}$
${\displaystyle \nabla \times {\textbf {H}}={\textbf {J}}+{\frac {\partial {\textbf {D}}}{\partial t}},}$
${\displaystyle \nabla \cdot {\textbf {B}}=0}$

where E is the electric field intensity, H is the magnetizing force, B is the magnetic field strength, D is the electric displacement, J is the current density, and ${\displaystyle \rho }$ the charge density. In the cgs system, 1/c (where c=velocity of light in a vacuum) precedes the time derivatives and ${\displaystyle 4\pi }$; ; precedes the J and ${\displaystyle \rho }$. These relations can also be expressed by an equivalent system of integral equations. In geophysical applications, it is normal to assume these fields are related by linear constituent equations:

Ohm's Law

${\displaystyle {\textbf {J}}=\sigma {\textbf {E}}}$

${\displaystyle {\textbf {D}}=\epsilon {\textbf {E}}}$

${\displaystyle {\textbf {B}}=\mu {\textbf {H}}}$

The last equation is sometimes written as

${\displaystyle {\textbf {B}}=\mu \mu _{0}{\textbf {H}}}$

where ${\displaystyle \sigma }$ is the conductivity, ${\displaystyle \epsilon }$ the dielectric permittivity, ${\displaystyle \mu }$ the magnetic permeability, and ${\displaystyle \mu _{0}}$ the permeability of free space.

Developed by James Clerk Maxwell (1834-1879), English physicist.