Maxwell equations

The Maxwell equations

There are four basic equations, called Maxwell equations, which form the axioms of electrodynamics. The so called local forms of these equations are the following:

rot H = j + ¶D/¶t                    (1)

rot E = - ¶B/¶t                        (2)

div B = 0                                 (3)

div D = r                                (4)

Here rot (or curl in English literature) is the so called vortex density, H is vector of the magnetic field strength, j is the current density vector, ¶D/¶t is the time derivative of the electric displacement vector D, E is the electric field strength, ¶B/¶t is the time derivative of the magnetic induction vector B, div is the so called source density and r is the charge density.

While the above local or differential forms are easy to remember and useful in applications, they are not so easy to understand as they use vector calculus to give spatial derivatives of vector fields like rot H or div D. The global or integral forms of the Maxwell equations are somewhat more complicated but at the same time they can be understood without knowing vector calculus. They are using path, surface, and volume integrals, however:

H^·dr = I + I^DISP                (1)

E^·dr = - ¶F_B/¶t                (2)

B^·dA = 0                           (3)

D^·dA =  rdV               (4)

where

I is the electric current            I = j^·dA,

I^DISP is the so called displacement       I ^DISP=  (¶D/¶t)^·dA,   and

F_B is the flux of the magnetic induction B   F_B = B^·dA.

It is important to realize that there are two variables to describe the electric properties of the electromagnetic field namely E and D, and also two variables for the magnetic properties of the field H and B. This is necessary when some materials are present with oriented electric and magnetic dipoles. If the electric dipole density is denoted by P, and the magnetic dipole density by M, then we can use the following definitions for D and B:

D = e₀E + P    and,

B = m₀H + M         .

Here e₀ and m₀ are the permittivity and the permeability of the vacuum, respectively. If we are in vacuum (P = 0, M =0 ) then the Maxwell equations can be written in the following form:

rot H = j + e₀¶E /¶t                (1)

rot E = - m₀¶H /¶t                   (2)

div H = 0                                 (3)

div E = r/e₀                             (4)

Thus we can see that in this case there are only one variable for the electric field E, and another variable H for the magnetic field. In other words the introduction of two more variables D and B (or P and M ) is necessary only if we have not only vacuum, but some material is also present. To determine j, P, and M for a certain material we use the so called material equations

j= j(E, Eⁱ),       P =  P (E),  and  M = M ( H ).

Here Eⁱ includes all non electromagnetic forces. The various functions in the material equations can be different for each material, but they are often linear. In that case the material equations are written in the following form:

j= s×(E+ Eⁱ),   P =  c_e×e₀×(E), and  M = c_m×m₀×( H ),

where s is the electric conductivity, c_e is the electric and c_m is the magnetic susceptibility.

            Thus the governing equations of electromagnetism include the 4 Maxwell equations and the 3 material equations. Finally one more equation is needed to establish a connection with mechanics e.g.

f = r×E + jxB

where f is the mechanical force density (force acting on the unit volume). Another possibility to establish the connection to mechanic is

r_EE = ½( E×D + H×B ),

where r_EE is the electromagnetic energy density, that is the energy stored by the electric and magnetic fields in the unit volume. (The concept of force and energy were developed already in mechanics.)

Classification of various chapters of electrodynamics

based on the Maxwell equations

The four Maxwell equations can be simplified omitting certain terms regarding the dynamics of the process.

Electro- and magnetostatics :

no current: j=0, no change in the magnetic induction ¶B/¶t = 0, no change in the electric inductions ¶D/¶t = 0.

Thus basic equations of electrostatics

div D = r,  and

rot E = 0         .

Equations of magnetostatics:

rot H = 0, and

div B = 0         .

As we can see in statics  there is no connection between the equations of electricity and magnetism.

Stationary fields (direct current):

We have already current j¹0, but the magnetic and the electric fields are not changing:

¶B/¶t = 0 and ¶D/¶t = 0.

In this case the Maxwell equations can be simplified to the following form:

rot H = j                                  (1)

rot E = 0                                 (2)

div B = 0                                 (3)

div D = r                                (4)

Quasi-stationary fields (e. g. alternating current):

We have electric current j¹0, and the magnetic field is changing ¶B/¶t ¹ 0, but the rate of changing o the  electric field can be neglected  ¶D/¶t » 0.

In this case the Maxwell equations can be simplified to the following form:

rot H = j                                  (1)

rot E = - ¶B/¶t                        (2)

div B = 0                                 (3)

div D = r                                (4)

Rapidly changing electromagnetic fields (electromagnetic waves):

In this case we have to regard all terms of the Maxwell equations. There is a special situation, however, when electromagnetic waves are propagating in vacuum. In this case the following simplifications can be applied:

no current: j=0 (no electron beam or other particles conducting current)

r = 0 (no charge in the space)

Thus the Maxwell equations for electromagnetic waves:

rot H = e₀¶E /¶t                      (1)

rot E = - m₀¶H /¶t                   (2)

div H = 0                                 (3)

div E = 0                                 (4)