Page 343 - Intro to Tensor Calculus

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where K e =1 + α e is the relative dielectric constant. The equation (2.6.45) are constitutive equations for
dielectric materials.
The effect of polarization is to produce regions of bound charges ρ b within the material and bound
surface charges µ b together with free charges ρ f which are not a result of the polarization. Within dielectrics
~
~
we have ∇· P = ρ b for bound volume charges and P · b e n = µ b for bound surface charges, where b e n is a
unit normal to the bounding surface of the volume. In these circumstances the expression for the potential
function is written ZZZ ZZ
1 ρ b 1 µ b
V = dτ + dσ (2.6.46)
4π 0 r 4π 0 r
V S
and the Gauss law becomes
~ ∗ ~ ~ ~
0 ∇· E = ρ = ρ b + ρ f = −∇ · P + ρ f or ∇( 0 E + P)= ρ f . (2.6.47)

~
~
~
Since D = 0E + P the Gauss law can also be written in the form
~ i
∇· D = ρ f or D = ρ f . (2.6.48)
,i
When no confusion arises we replace ρ f by ρ. In integral form the Gauss law for dielectrics is written
ZZ
~
D · ˆ n dσ = Q fe (2.6.49)
S
where Q fe is the total free charge density within the enclosing surface.

Magnetostatics
~
~
A stationary charge generates an electric field E while a moving charge generates a magnetic field B.
Magnetic field lines associated with a steady current moving in a wire form closed loops as illustrated in the
figure 2.6-5.

Figure 2.6-5. Magnetic field lines.

The direction of the magnetic force is determined by the right hand rule where the thumb of the right
hand points in the direction of the current flow and the fingers of the right hand curl around in the direction
~
~
of the magnetic field B. The force on a test charge Q moving with velocity V in a magnetic field is
~
~
~
F m = Q(V × B). (2.6.50)
The total electromagnetic force acting on Q is the electric force plus the magnetic force and is
h i
~
~
~
~
F = Q E +(V × B) (2.6.51)