Introductory Physics Volume Two

7.1 Maxwell Equations 137
7 Wave Optics
§ 7.1 Maxwell Equations
Let us write down, in one place, all of the fundamental equations
about the electric and magnetic fields.
Gauss ′ s Law
Faraday ′ s Law
Ampere ′ s Law
Gauss ′ s Law for B
∮
∮
∮
∮
⃗E · dA ⃗ = 1 ∫
ɛ 0
⃗E · ⃗dl = − d dt
∫
⃗B · ⃗dl = µ 0
⃗B · ⃗ dA = 0
ρ dV
∫
⃗B · ⃗ dA
⃗J · ⃗ dA + µ 0 ɛ 0
d
dt
∫
⃗E · ⃗ dA
These equations as a group are known as the Maxwell Equations.
The last equation, which we have not discussed before, can be
understood to say that there is no magnetic equivalent to the electric
charge. That is, that there are no magnetic charges, places where
magnetic field lines begin or end, thus that magnetic field lines do not
end. If you want to think of the magnetic field as a flowing fluid, with
B the velocity of the fluid, then this law says that the fluid does not
compress: B flows like water, when it spreads out the velocity decreases,
when it goes through a narrow region it speeds up, the volume rate of
flow is a constant.
Ok, so the new law tells us that there are no magnetic charge, let
us see what the Maxwell equations look like in a region where there is
no electric charge (ρ = 0) and no current (J = 0).