Page 35 - Electromagnetics

In Minkowski’s form of Maxwell’s equations, the mediating field is the electromagnetic
field consisting of the set of four vector fields E(r, t), D(r, t), B(r, t), and H(r, t). The field
equations are the four partial differential equations referred to as the Maxwell–Minkowski
equations
∂
∇× E(r, t) =− B(r, t), (2.1)
∂t
∂
∇× H(r, t) = J(r, t) + D(r, t), (2.2)
∂t
∇· D(r, t) = ρ(r, t), (2.3)
∇· B(r, t) = 0, (2.4)
along with the continuity equation

∂
∇· J(r, t) =− ρ(r, t). (2.5)
∂t
Here (2.1) is called Faraday’s law, (2.2) is called Ampere’s law, (2.3) is called Gauss’s
law, and (2.4) is called the magnetic Gauss’s law. For brevity we shall often leave the
dependence on r and t implicit, and refer to the Maxwell–Minkowski equations as simply
the “Maxwell equations,” or “Maxwell’s equations.”
Equations (2.1)–(2.5), the point forms of the field equations, describe the relation-
ships between the fields and their sources at each point in space where the fields are
continuously differentiable (i.e., the derivatives exist and are continuous). Such points
are called ordinary points. We shall not attempt to define the fields at other points,
but instead seek conditions relating the fields across surfaces containing these points.
Normally this is necessary on surfaces across which either sources or material parameters
are discontinuous.
The electromagnetic fields carry SI units as follows: E is measured in Volts per meter
(V/m), B is measured in Teslas (T), H is measured in Amperes per meter (A/m), and
2
D is measured in Coulombs per square meter (C/m ). In older texts we find the units of
2
B given as Webers per square meter (Wb/m ) to reflect the role of B as a flux vector; in
2
that case the Weber (Wb = T·m ) is regarded as a unit of magnetic flux.
The interdependence of Maxwell’s equations. It is often claimed that the diver-
gence equations (2.3) and (2.4) may be derived from the curl equations (2.1) and (2.2).
While this is true, it is not proper to say that only the two curl equations are required
to describe Maxwell’s theory. This is because an additional physical assumption, not
present in the two curl equations, is required to complete the derivation. Either the
divergence equations must be specified, or the values of certain constants that fix the
initial conditions on the fields must be specified. It is customary to specify the divergence
equations and include them with the curl equations to form the complete set we now call
“Maxwell’s equations.”
To identify the interdependence we take the divergence of (2.1) to get

∂B
∇· (∇× E) =∇ · − ,
∂t
hence
∂
(∇· B) = 0
∂t

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