BASICS OF MAXWELL'S LAWS
All induction logs and electromagnetic propagation logs are designed upon the principles of Maxwell's Laws of electromagnetism. You don't need to know the math or the theory to use the log data, but it's here if you want it. Here is a brief summary:
 

Although I learned all this in University 50 years ago, I can no longer recite any of it from memory. Therefore, the following is from Wikipedia http://en.wikipedia.org/wiki/Maxwell's_laws , with a little editing. If you don't have strong calculus skills, most of this material will be meaningless. Have fun!

Definitions and units
Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities. The equations in this section, unless otherwise stated, are given in SI units. Unlike the equations of mechanics, Maxwell's equations are changed in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places.
Symbol Meaning (first term is the most common) SI Unit of Measure
the divergence operator per meter (factor contributed by applying either operator)
the curl operator
partial derivative with respect to time per second (factor contributed by applying the operator)
electric field volt per meter or, equivalently,
newton per coulomb
magnetic field
also called the magnetic induction
also called the magnetic field density
also called the magnetic flux density
tesla, or equivalently,
weber per square meter
volt•second per square meter
electric displacement field coulombs per square meter or, equivalently,
newton per volt-meter
magnetizing field
also called auxiliary magnetic field
also called magnetic field intensity
also called magnetic field
ampere per meter
permittivity of free space, officially the electric constant,
a universal constant
farads per meter
magnetic permeability of free space, officially the magnetic constant,
a universal constant
henries per meter, or newtons per ampere squared
free charge density (not including bound charge) coulomb per cubic meter
total charge density (including both free and bound charge) coulomb per cubic meter
the flux of the electric field over any closed gaussian surface S joule-meter per coulomb
net unbalanced free electric charge enclosed by the
Gaussian surface S (not including bound charge)
coulombs
net unbalanced electric charge enclosed by the Gaussian
surface S (including both free and bound charge)
coulombs
the flux of the magnetic field over any closed surface S tesla meter-squared or weber
line integral of the electric field along the boundary ∂S
(therefore necessarily a closed curve) of the surface S
joule per coulomb
magnetic flux over any surface S (not necessarily closed) weber
free current density (not including bound current) ampere per square meter
total current density (including both free and bound current) ampere per square meter
line integral of the magnetic field over
the closed boundary ∂S of the surface S
tesla-meter
net free electrical current passing through
the surface S (not including bound current)
amperes
net electrical current passing through the
surface S (including both free and bound current)
amperes
electric flux through any surface S, not necessarily closed joule-meter per coulomb
flux of electric displacement field through any surface S, not necessarily closed coulombs
differential vector element of surface area A, with infinitesimally
 

small magnitude and direction normal to surface S

square meters
differential vector element of path length tangential to contour meters

MODERN VERSIONS OF MAXWELL'S LAWS
Two equivalent, general formulations of Maxwell's equations follow. The first separates free charge and free current from bound charge and bound current. This separation is useful for calculations involving dielectric and/or magnetized materials. The second formulation treats all charge equally, combining free and bound charge into total charge (and likewise with current). Of course, such an approach applies where no dielectric or magnetic material is present, and therefore no bound charge or current, but it also is a more fundamental or microscopic point of view.

Table 1: Formulation in terms of free charge and current
Name Differential form Integral form
Gauss's law:
Gauss's law for magnetism:
Maxwell-Faraday equation
(Faraday's law of induction):
Ampère's Circuital Law
(with Maxwell's correction):

 

Table 2: Formulation in terms of total charge and current
Name Differential form Integral form
Gauss's law:
Gauss's law for magnetism:
Maxwell-Faraday equation
(Faraday's law of induction):
Ampère's Circuital Law
(with Maxwell's correction):

Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the magnetic permittivity and magnetic permeability of a material. At the microscopic level, Maxwell's equations, ignoring quantum effects, describe fields, charges and currents in free space — but at this level of detail one must include all charges, even those at an atomic level, generally an intractable problem.

 

SPECIAL CASES
Bound charge, and proof that formulations are equivalent

If an electric field is applied to a dielectric material, each of the molecules responds by forming a microscopic dipole -- its atomic nucleus will move a tiny distance in the direction of the field, while its electrons will move a tiny distance in the opposite direction. This is called polarization of the material. The distribution of charge that results from these tiny movements turn out to be identical to having a layer of positive charge on one side of the material, and a layer of negative charge on the other side -- a macroscopic separation of charge, even though all of the charges involved are "bound" to a single molecule. This is called bound charge. Likewise, in a magnetized material, there is effectively a "bound current" circulating around the material, despite the fact that no individual charge is traveling a distance larger than a single molecule. The relation between polarization, magnetization, bound charge, and bound current is as follows:

where P and M are polarization and magnetization, and ρb and Jb are bound charge and current, respectively. Plugging in these relations, it can be easily demonstrated that the two formulations of Maxwell's equations given above are precisely equivalent.

Constitutive Relations
In order to apply Maxwell's equations (the formulation in terms of free charge and current, and D and H), it is necessary to specify the relations between D and E, and H and B. These are called constitutive relations, and correspond physically to specifying the response of bound charge and current to the field, or equivalently, how much polarization and magnetization a material acquires in the presence of electromagnetic fields.

Case without magnetic or dielectric materials
In the absence of magnetic or dielectric materials, the relations are simple:

where ε0 and μ0 are two universal constants, called the permittivity of free space and permeability of free space, respectively.

Case of linear materials
In a "linear", isotropic, nondispersive, uniform material, the relations are also straightforward:

where ε and μ are constants (which depend on the material), called the permittivity and permeability, respectively, of the material.

General case
For real-world materials, the constitutive relations are not simple proportionalities, except approximately. The relations can usually still be written:

but ε and μ are not, in general, simple constants, but rather functions. For example, ε and μ can depend upon:

  • The strength of the fields (the case of nonlinearity, which occurs when ε and μ are functions of E and B; see, for example, Kerr and Pockels effects),
     
  • The direction of the fields (the case of anisotropy, birefringence, or dichroism; which occurs when ε and μ are second-rank tensors),
     
  • The frequency with which the fields vary (the case of dispersion, which occurs when ε and μ are functions of frequency; see, for example, Kramers–Kronig relations),
     
  • The position inside the material (the case of a nonuniform material, which occurs when ε and μ vary from point to point within the material; for example in a domained structure, heterostructure or a liquid crystal),
     
  • The history of the fields (the case of hysteresis, which occurs when ε and μ are functions of both present and past values of the fields).

Equations in terms of E and B for linear materials
Substituting in the constitutive relations above, Maxwell's equations in a linear material (differential form only) are:

These are formally identical to the general formulation in terms of E and B (given above), except that the permittivity of free space was replaced with the permittivity of the material (see also displacement field, electric susceptibility and polarization density), the permeability of free space was replaced with the permeability of the material, and only free charges and currents are included (instead of all charges and currents).

Maxwell's equations in vacuum
Starting with the equations appropriate in the case without dielectric or magnetic materials, and assuming that there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:

These equations have a solution in terms of traveling sinusoidal plane waves, with the electric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields in phase, traveling at the speed.

The traveling wave solution is found by substitution of one of the curl equations into the time derivative of the other, producing:

which reduces to the electromagnetic wave equation due to an identity in vector calculus. The equation is satisfied in one dimension, for example, by a solution of the form E = E( x − c0t ), that is, by a solution that is unchanged when t advances to t + Δt at a position x that advances to x + c0 Δt.

Maxwell discovered that this quantity c0 is the speed of light in vacuum, and thus that light is a form of electromagnetic radiation. The current SI values for the speed of light, the electric and the magnetic constant are summarized in the following table.

Symbol Name Numerical Value SI Unit of Measure Type
Speed of light in vacuum 299 792.458 meters per second defined
Electric constant 8. 854 187 817
  x 10^-12
Farads per meter derived  
Magnetic constant 1.2566 x 10^-6 Henrys per meter defined

 

Nondimensionalization and unobservability of the speed of light
Because c0 and μ0 have defined values (they are properties of the ideal reference state of free space), they are not subject to alteration due to experimental observation. For example, if length is measured in units λ and time in units τ, the distance x in units of λ becomes x = λ ζ and the time t becomes t = τ η, where ζ is the number of length units in x and η is the number of time units in t. The above curl equation for the traveling wave becomes:

and because the SI units are related by λ = c0τ this equation does not depend any longer on the speed of light. Experiment could in principle, however, alter the standard meter, for example, as a result of greater

measurement accuracy.

With magnetic monopoles
Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The standard form of the equations provide for an electric charge, but posit no magnetic charge. Except for this, the equations are symmetric under interchange of electric and magnetic field. In fact, symmetric equations can be written when all charges are zero, and this is how the wave equation is derived .

Fully symmetric equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges. With the inclusion of a variable for these magnetic charges, there will also be "magnetic current" variable in the equations. The extended Maxwell's equations, simplified by nondimensionalization, are as follows:

Name Without Magnetic Monopoles With Magnetic Monopoles (hypothetical)
Gauss's law:
Gauss' law for magnetism:
Maxwell-Faraday equation
(Faraday's law of induction):
Ampère's law
(with Maxwell's extension):
       

If magnetic charges do not exist, or if they exist but where they are not present in a region, then the new variables are zero, and the symmetric equations reduce to the conventional equations of electromagnetism such as . Classically, the question is "Why does the magnetic charge always seem to be zero?"

Materials and dynamics
The fields in Maxwell's equations are generated by charges and currents. Conversely, the charges and currents are affected by the fields through the Lorentz force equation:

where q is the charge on the particle and v is the particle velocity. (It also should be remembered that the Lorentz force is not the only force exerted upon charged bodies, which also may be subject to gravitational, nuclear, etc. forces.) Therefore, in both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics.  This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents, which enter Maxwell's equations through the constitutive equations, as described next.

Commonly, real materials are approximated as "continuum" media with bulk properties such as the refractive index, permittivity, permeability, conductivity, and/or various susceptibilities. These lead to the macroscopic Maxwell's equations, which are written (as given above) in terms of free charge/current densities and D, H, E, and B ( rather than E and B alone ) along with the constitutive equations relating these fields. For example, although a real material consists of atoms whose electronic charge densities can be individually polarized by an applied field, for most purposes behavior at the atomic scale is not relevant and the material is approximated by an overall polarization density related to the applied field by an electric susceptibility.

Continuum approximations of atomic-scale inhomogeneities cannot be determined from Maxwell's equations alone. but require some type of quantum mechanical analysis such as quantum field theory as applied to condensed matter physics. See, for example, density functional theory, Green–Kubo relations and Green's function (many-body theory). Various approximate transport equations have evolved, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier-Stokes equations. Some examples where these equations are applied are magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, plasma modeling. An entire physical apparatus for dealing with these matters has developed. A different set of homogenization methods (evolving from a tradition in treating materials such as conglomerates and laminates) are based upon approximation of an inhomogeneous material by a homogeneous effective medium (valid for excitations with wavelengths much larger than the scale of the inhomogeneity).

 

Real world issues
Theoretical results have their place, but often require fitting to experiment. Continuum-approximation properties of many real materials rely upon measurement, for example, ellipsometry measurements.

In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant where frequency is limited to a narrow bandwidth; material absorption can be neglected for wavelengths where a material is transparent; and metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero skin depth of field penetration).

And, of course, some situations demand that Maxwell's equations and the Lorentz force be combined with other forces that are not electromagnetic. An obvious example is gravity. A more subtle example, which applies where electrical forces are weakened due to charge balance in a solid or a molecule, is the Casimir force from quantum electrodynamics.

The connection of Maxwell's equations to the rest of the physical world is via the fundamental sources of charges and currents and the forces on them, and also by the properties of physical materials.

Boundary conditions
Although Maxwell's equations apply throughout space and time, practical problems are finite and solutions to Maxwell's equations inside the solution region are joined to the remainder of the universe through boundary conditions and started in time using initial conditions. In some cases, like waveguides or cavity resonators, the solution region is largely isolated from the universe, for example, by metallic walls, and boundary conditions at the walls define the fields with influence of the outside world confined to the input/output ends of the structure. In other cases, the universe at large sometimes is approximated by an artificial absorbing boundary, or, for example for radiating antennas or communication satellites, these boundary conditions can take the form of asymptotic limits imposed upon the solution.

In addition, for example in an optical fiber or thin-film optics, the solution region often is broken up into subregions with their own simplified properties, and the solutions in each subregion must be joined to each other across the subregion interfaces using boundary conditions. Following are some links of a general nature concerning boundary value problems: Examples of boundary value problems, Sturm-Liouville theory, Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, Cauchy boundary condition, Sommerfeld radiation condition. Needless to say, one must choose the boundary conditions appropriate to the problem being solved.

THE HEAVISIDE VERSIONS
Gauss's law
describes the relation between the electric field and the distribution of electric charge, as follows:

The formulation of Table 1 is assumed; that is, ρf is the "free" electric charge density (in units of C/m³), not including bound charge from the polarization of a material, and is the electric displacement field (in units of C/m²). For stationary charges in vacuum, the force exerted upon one point charge by another as found from Gauss's law is Coulomb's law.

The equivalent integral form (by the divergence theorem) of Gauss' law is:

where:

S is any fixed, closed surface,
The integral is a surface integral, i.e. is a vector whose magnitude is the area of a differential square on the closed surface A, and whose direction is an outward-facing normal vector, and
Qenclosed is the free charge enclosed within the surface S. (If the surface itself is charged, that gives an extra contribution weighted by a factor 1/2.)

In a linear, isotropic, homogeneous material, with instantaneous response to field changes, D is directly related to the electric field E  via a material-dependent constant called the permittivity, ε:

.

The material permittivity ε can also be written as ε0 εr where εr is the material's relative permittivity or its dielectric constant. No material (except free space) is precisely linear and isotropic, but many materials are approximately so. The permittivity of free space, or electric constant, is denoted as ε0 (approximately 8.854 pF/m), and appears in:

where, again,  E  is the electric field (in units of V/m), ρt is the total charge density (including bound charges). The formulation of Table 2 is assumed.

Some insight into Gauss' law is found using the Maxwell-Faraday equation:

which shows the solenoidal portion of E is determined by the time variation of the magnetic field. Thus, in electrostatics (that is, when the system is unchanging in time), by Helmholtz decomposition the E-field can be expressed in terms of a scalar field as:

Time independence not only allows E to be expressed as a gradient, but also removes any time-delay in material response (ε independent of time), so the equation determining the electrostatic potential ɸ (r ) is:

which is Poisson's equation in the case where ε is independent of position (that is, when the material is homogeneous). The formulation of Table 1 is assumed. That is, the bound charge is subsumed under the permittivity, and only the free charge is explicit on the right side of the equation.

Gauss's law for magnetism states that the divergence of the magnetic field is always zero (in other words, the magnetic field is a solenoidal vector field):

where is the magnetic B-field (in units of tesla, denoted "T"), also called "magnetic flux density", "magnetic induction", or simply "magnetic field". It is interpreted as saying there is no "magnetic" charge that is the analog of the electric charge, and often this equation is referred to as "the absence of magnetic monopoles". Differently put, the basic entity for magnetism is the magnetic dipole, which orients itself in a magnetic field.

By the divergence theorem, the above divergence equation has an equivalent integral form:

where is an infinitesimal vector corresponding to the area of a differential square on the surface S with an outward facing surface normal defining its direction.

Like the electric field's integral form, this equation works only if the integral is done over a closed surface.

This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, the above reference to this law as saying there are no magnetic monopoles.

The Maxwell-Faraday equation states:

This equation is usually referred to as "Faraday's law of induction", but in fact it is only a restricted form of Faraday's law; for example, it doesn't apply to situations involving motionally induced EMF.

Ampère's circuital law describes the source of the magnetic field,

where is the magnetic field strength (in units of A/m), related to the magnetic flux density by a constant called the permeability, μ (), and J is the current density, defined by: where is a vector field called the drift velocity that describes the velocities of the charge carriers which have a density described by the scalar function ρq. The second term on the right hand side of Ampère's Circuital Law is known as the displacement current.

It was Maxwell who added the displacement current term to Ampère's Circuital Law at equation (112) in his 1861 paper On Physical Lines of Force.

Maxwell used the displacement current in conjunction with the original eight equations in his 1865 paper A Dynamical Theory of the Electromagnetic Field to derive a wave equation that has the velocity of light. Most modern textbooks derive this electromagnetic wave equation using the 'Heaviside Four'.

In free space, the permeability μ is the magnetic constant, μ0, which is defined to be exactly 4π×10-7 Wb/A•m. Also, the permittivity becomes the electric constant ε0, also a defined quantity. Thus, in free space, the equation becomes:

Using Stokes theorem the equivalent integral form can be found:

C is the edge of the open surface A (any surface with the curve C as its edge will do), and Iencircled is the current encircled by the curve C (the current through any surface is defined by the equation: ). Sometimes this integral form of Ampere-Maxwell Law is written as:

    because the term    

is displacement current. The displacement current concept was Maxwell's greatest innovation in electromagnetic theory. It implies that a magnetic field appears during the charge or discharge of a capacitor. If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

 

MAXWELL'S EQUATIONS IN CGS UNITS
The above equations are given in the International System of Units, or SI for short. In a related unit system, called cgs (short for centimeter-gram-second), the equations take the following form:

Where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become:

In this system of units the relation between magnetic induction, magnetic field and total magnetization take the form:

With the linear approximation:

χm for vacuum is zero and therefore:

and in the ferro or ferri magnetic materials where χm is much bigger than 1:

The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

where is the charge on the particle and is the particle velocity. This is slightly different from the SI-unit
expression above. For example, here the magnetic field has the same units as the electric field .

Maxwell's equations and special relativity
Maxwell's equations have a close relation to special relativity: Not only were Maxwell's equations a crucial part of the historical development of special relativity, but also, special relativity has motivated a compact mathematical formulation Maxwell's equations, in terms of covariant tensors.

 

MAXWELL'S ORIGINAL EQUATIONS
The eight original Maxwell's equations can be written in modern vector notation as follows:

(A) The law of total currents
(B) The equation of magnetic force
(C) Ampère's circuital law
(D) Electromotive force created by convection, induction, and by static electricity. (This is in effect the Lorentz force)
(E) The electric elasticity equation
(F) Ohm's law
(G) Gauss's law
(H) Equation of continuity
Notation
is the magnetizing field, which Maxwell called the "magnetic intensity".
is the electric current density (with being the total current including displacement current).
is the displacement field (called the "electric displacement" by Maxwell).
ρ is the free charge density (called the "quantity of free electricity" by Maxwell).
is the magnetic vector potential (called the "angular impulse" by Maxwell).
is called the "electromotive force" by Maxwell. The term electromotive force is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term electric field.
Φ is the electric potential (which Maxwell also called "electric potential").
σ is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

It is interesting to note the term that appears in equation D. Equation D is therefore effectively the Lorentz force, similarly to equation (77) of his 1861 paper (see above).

When Maxwell derives the electromagnetic wave equation in his 1865 paper, he uses equation D to cater for electromagnetic induction rather than Faraday's law of induction which is used in modern textbooks. (Faraday's law itself does not appear among his equations.) However, Maxwell drops the term from equation D when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.

 
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