Themes > Science > Physics > Electromagnetism > Magnetostatics > Magnetic Field > Maxwell's equations in differential form

Maxwell's equations (in vacuum ) consist of Gauss' law for the electric field ${\bf E}$,

 
 \begin{displaymath} \bigtriangledown . {\bf E} = 4 \pi k \rho ,\end{displaymath} (103)

Gauss' law for the magnetic field ${\bf B}$, 
 \begin{displaymath} \bigtriangledown . {\bf B} = 0 ,\end{displaymath} (104)

Faraday's law  
 \begin{displaymath} \bigtriangledown \times {\bf E} + g \frac{\partial {\bf B}}{\partial t} =0\end{displaymath} (105)

and Ampère's law  
 \begin{displaymath} \bigtriangledown \times {\bf B} - \frac{1}{g c^2 } \frac{\partial {\bf E}}{\partial t} = \frac{4\pi k}{g c^2 } {\bf j}\end{displaymath} (106)

where $\rho $ is the charge density (charge per unit volume $\rho \equiv \frac{dq}{d\tau }$) and ${\bf j}$ is the current density (charge per unit area ${\bf j} \equiv \frac{di}{dA} {\bf \hat {A}}$). k and g are constants and c is the speed of light in vacuum.

The Lorentz force law is  
 \begin{displaymath} {\bf F} = q ( {\bf E} + g {\bf v} \times {\bf B})\end{displaymath} (107)

which gives the force ${\bf F}$ on a particle of charge q moving with velocity ${\bf v}$ in an electromagnetic field.

Later we shall see that the constant k is the same one that appears in Coulomb's law for the electric force between two point charges  
 \begin{displaymath} {\bf F} = k \frac{q_{1} q_{2}}{r^2} \hat {r}\end{displaymath} (108)

and the constant g specifies the relative strength of the ${\bf E}$ and ${\bf B}$ fields. An excellent and more complete discussion of units may be found in the book by Jackson. In terms of Jackson's constants (k1 and k3) the relation is k=k1 and g=k3. From Coulomb's law it can be seen that the units chosen for charge and length will determine the units for k. The three main systems of units in use are called Heaviside-Lorents, CGS or Gaussian and MKS or SI. The values of the constants in these unit systems are specified in Table 3.1 below.

  Heaviside-Lorentz CGS (Gaussian) SI (MKS)
k $\frac{1}{4\pi}$ 1 $\frac{1}{4\pi {\epsilon}_{0}}$
g $\frac{1}{c}$ $\frac{1}{c}$ 1

Table 3.1

Inserting these constants into Maxwell's equations and the Lorentz force law gives the equations as they appear in different unit systems. In Heaviside-Lorentz units Maxwell's equations are  
 \begin{displaymath} \bigtriangledown . {\bf E} = \rho\end{displaymath} (109)
 
 \begin{displaymath} \bigtriangledown . {\bf B} = 0\end{displaymath} (110)
 
 \begin{displaymath} \bigtriangledown \times {\bf E} + \frac{1}{c} \frac{\partial {\bf B}}{\partial t} =0\end{displaymath} (111)
 
 \begin{displaymath} \bigtriangledown \times {\bf B} - \frac{1}{c} \frac{\partial {\bf E}}{\partial t} = \frac{1}{c} {\bf j}\end{displaymath} (112)

and the Lorentz force law is  
 \begin{displaymath} {\bf F} = q ( {\bf E} + \frac{1}{c} {\bf v} \times {\bf B}) .\end{displaymath} (113)

In CGS or Gaussian units Maxwell's equations are  
 \begin{displaymath} \bigtriangledown . {\bf E} = 4 \pi \rho\end{displaymath} (114)
 
 \begin{displaymath} \bigtriangledown . {\bf B} = 0\end{displaymath} (115)
 
 \begin{displaymath} \bigtriangledown \times {\bf E} + \frac{1}{c} \frac{\partial {\bf B}}{\partial t} =0\end{displaymath} (116)
 
 \begin{displaymath} \bigtriangledown \times {\bf B} - \frac{1}{c} \frac{\partial {\bf E}}{\partial t} = \frac{4\pi }{c} {\bf j}\end{displaymath} (117)
and the Lorentz force law is  
 \begin{displaymath} {\bf F} = q ( {\bf E} + \frac{1}{c} {\bf v} \times {\bf B}) .\end{displaymath} (118)

In MKS or SI units Maxwell's equations are  
 \begin{displaymath} \bigtriangledown . {\bf E} = \frac{\rho }{{\epsilon}_{0}}\end{displaymath} (119)
 
 \begin{displaymath} \bigtriangledown . {\bf B} = 0\end{displaymath} (120)
 
 \begin{displaymath} \bigtriangledown \times {\bf E} + \frac{\partial {\bf B}}{\partial t} =0\end{displaymath} (121)
 
 \begin{displaymath} \bigtriangledown \times {\bf B} - \frac{1}{c^2 } \frac{\partial {\bf E}}{\partial t} = \frac{1}{c^2 {\epsilon}_{0}} {\bf j}\end{displaymath} (122)

However, because $c^2 = \frac{1}{{\mu}_{0}{\epsilon}_{0}}$ (see Section x.x) in SI units this last equation is usually written  
 \begin{displaymath} \bigtriangledown \times {\bf B} - {\mu}_{0}{\epsilon}_{0} \frac{\partial {\bf E}}{\partial t} = {\mu}_{0} {\bf j}\end{displaymath} (123)

and the Lorentz force law is  
 \begin{displaymath} {\bf F} = q ( {\bf E} + {\bf v} \times {\bf B}) .\end{displaymath} (124)

Particle physicists most often use Heaviside-Lorentz units and furthermore usually use units in which $c \equiv 1$, so that Maxwell's equations are in their simplest possible form. CGS units are used in the books by Jackson  and Ohanian  and Marion , whereas MKS units are used by Griffiths  and most freshman physics texts. In this book we shall use Maxwell's equations as presented in equations  so that all of our equations will contain the constants k and n. The equations for a specific unit system can then simply be obtained by use of Table 3.1. Thus this book will not make a specific choice of units. The advantage of this is that comparison of results in this book with the references will be made much easier.


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