electrodynamics / Electromagnetic Field Theory - Bo Thide

allowing us to rewrite Equation (1.32) on the facing page in the following way:

where EEMF is the field which is induced in the “loop,” i.e., in the moving system. The use of Stokes' theorem “backwards” on Equation (1.37) above yields

Hence, a moving observer measures the following Lorentz force on a charge q

corresponding to an “effective” electric field in the “loop” (moving observer)

Hence, we can conclude that for a stationary observer, the Maxwell equation

is indeed valid even if the “loop” is moving.

1.3.5 Maxwell's microscopic equations

We are now able to collect the results from the above considerations and formulate the equations of classical electrodynamics valid for arbitrary variations in time and space of the coupled electric and magnetic fields E(t;x) and B(t;x). The equations are

In these equations (t;x) represents the total, possibly both time and space dependent, electric charge, i.e., free as well as induced (polarisation) charges, and j(t;x) represents the total, possibly both time and space dependent, electric current, i.e., conduction currents (motion of free charges) as well as all atomistic (polarisation, magnetisation) currents. As they stand, the equations therefore incorporate the classical interaction between all electric charges and currents in the system and are called Maxwell's microscopic equations. Another name often used for them is the Maxwell-Lorentz equations. Together with the appropriate constitutive relations, which relate and j to the fields, and the initial and boundary conditions pertinent to the physical situation at hand, they form a system of well-posed partial differential equations which completely determine E and B.

1.3.6 Maxwell's macroscopic equations

The microscopic field equations (1.43) provide a correct classical picture for arbitrary field and source distributions, including both microscopic and macroscopic scales. However, for macroscopic substances it is sometimes convenient to introduce new derived fields which represent the electric and magnetic fields in which, in an average sense, the material properties of the substances are already included. These fields are the electric displacement D and the magnetising field H. In the most general case, these derived fields are complicated nonlocal, nonlinear functionals of the primary fields E and B:

Under certain conditions, for instance for very low field strengths, we may assume that the response of a substance is linear so that

i.e., that the derived fields are linearly proportional to the primary fields and that the electric displacement (magnetising field) is only dependent on the electric (magnetic) field.

The field equations expressed in terms of the derived field quantities D and H are

1.4 Electromagnetic Duality

If we look more closely at the microscopic Maxwell equations (1.48), we see that they exhibit a certain, albeit not a complete, symmetry. Let us for explicitness denote the electric charge density = (t;x) by e and the electric current density j = j(t;x) by je. We further make the ad hoc assumption that there exist magnetic monopoles represented by a magnetic charge density, denoted

m = m(t;x), and a magnetic current density, denoted jm =jm(t;x). With these new quantities included in the theory, the Maxwell equations can be written

We shall call these equations the Dirac-Maxwell equations or the electromagnetodynamic equations

where we used the fact that, according to formula (M.82) on page 175, the divergence of a curl always vanishes. Using (1.48c) to rewrite this relation, we obtain the equation of continuity for magnetic monopoles

which has the same form as that for the electric monopoles (electric charges) and currents, Equation (1.21) on page 9.

We notice that the new Equations (1.48) on the preceding page exhibit the following symmetry (recall that "0 0 = 1=c2):

which is a particular case ( = =2) of the general duality transformation (depicted by the Hodge star operator)

which leaves the Dirac-Maxwell equations, and hence the physics they describe (often referred to as electromagnetodynamics), invariant. Since E and je are (true or polar) vectors, B a pseudovector (axial vector), e a (true) scalar, then m and , which behaves as a mixing angle in a two-dimensional “charge space,” must be pseudoscalars and jm a pseudovector.

END OF EXAMPLE 1.1

Hence, a fixed mixing angle, or, equivalently, a fixed ratio between the electric and magnetic charges/currents, “hides” the magnetic monopole influence ( m and jm) on the dynamic equations.

We notice that the inverse of the transformation given by Equation (1.52) on page 16

Furthermore, from the expressions for the transformed charges and currents above, we find that

Furthermore, assuming that = (t;x), we see that the spatial and temporal differenti-

END OF EXAMPLE 1.4

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