Stigmatic Optics. Rafael G González-Acuña
of charge divided by the permittivity free space.
Here we limit ourselves to present this form of the Gauss law because the derivation of the divergence theorem is beyond the scope of the book. For a more detailed analysis of the divergence theorem, the reader is invited to read the references presented in the bibliography of this chapter.
1.5 The Gauss law for magnetism
Gauss’s law for magnetism has the same structure as the Gauss law of the previous section with the condition that there are no magnetic charges.
So, we start with the definition of magnetic flux through a surface S,
ΦB⃗=∫SB⃗·n⃗da,(1.10)
the magnetic field, B⃗, multiplied by the component of the area perpendicular to the field, where n⃗ is the unit normal vector of infinitesimal area da.
Therefore, over a closed surface S, Gauss’s law of magnetism is given by
∮SB⃗·n⃗da=0,(1.11)
As we mentioned in the introduction, there are no magnetic charges. What Gauss’s law of magnetism tells us is that the total magnetic flux passing through any closed surface is zero.
Tacking the knowledge acquired in the last section, we can get the vector form of Gauss’s law of magnetism, hence,
∇·B⃗=0.(1.12)
This happens, as expected because there are no magnetic charges. Therefore, the density of magnetic charge is zero.
1.6 Faraday’s law
Faraday’s electromagnetic induction law establishes that the electromotive force induced in a closed circuit is directly proportional to the speed with which the magnetic flux passing through any surface with the circuit as edge changes in time. Thus,
∮cE⃗·dl⃗=−ddt∫SB⃗·n⃗da,(1.13)
where E⃗ is the electric field, dl⃗ is the infinitesimal element of the length of the circuit represented by contour C, B⃗ is the magnetic field and S is an arbitrary surface, whose edge is C. The right-hand rule gives the directions of contour C and n⃗da.
The electromotive force or induced voltage (represented by emf) is any cause capable of maintaining a potential difference between two points in an open circuit or of producing an electric current in a closed circuit.
According to the Stokes theorem, the differential form of Faraday’s law is generally written as,
∇×E⃗=−∂B⃗∂t,(1.14)
where ∇×E⃗ is the curl of the electric field E⃗. The curl operates on a vector field and provides a vector result that designates the tendency of the field to circulate around a point and the direction of the axis of greatest circulation.
What tells us the differential form of Faraday’s law is that a circulating electric field is produced by a magnetic field that changes with time.
1.7 Ampère’s law
Ampère’s law, also called the Ampère–Maxwell law, is generally written in its integral form as,
∮cB⃗·dl⃗=μ0Ienc+ε0ddt∫SE⃗·n⃗da.(1.15)
The left side of equation (1.15) tells us about the circulation of the magnetic field around a closed path C. On the right side, we have two elements that originate the magnetic field. The first one is a steady current given by Ienc. The other one is the change in time of the electric flux through a surface bounded by C.
Please notice that in equation (1.15) the factor μ0 is a constant called the magnetic permeability of free space. In the international system of units, where force is in newtons (N) and current in amperes (A),
μ0=1.2566370614×10−6NA−2.(1.16)
Well, what equation (1.15) tells us is that an electric current or a changing electric flux through a surface produces a circulating magnetic field around any path that bounds that surface.
Now due the Stokes theorem, we can express the Ampère–Maxwell law as its differential form,
∇×B⃗=μ0J⃗+ε0∂E⃗∂t,(1.17)
The left side of the equation (1.17) is the circulating magnetic field. On the right side are the sources of the circulating magnetic field. Notice that the first term in the right side of equation (1.17), J⃗ is the current density vector. The second term on the right side of the mentioned equation is the rate of change of the electric field with time.
Therefore, what the Ampère–Maxwell law in its differential form tells us is that a circulating magnetic field is produced by an electric current and by an electric field that changes with time.
1.8 The wave equation
We have briefly reviewed the Maxwell equations, but enough that from them, we can obtain the wave equation. So we recall the set of Maxwell equations, equations (1.8), (1.12) (1.14) and (1.17), as equation (1.18),
∇·E⃗=ρε0,∇·B⃗=0,∇×E⃗=−∂B⃗∂t,∇×B⃗=μ0J⃗+ε0∂E⃗∂t,(1.18)
If we apply the curl on Faraday’s law, equation (1.14), we get,
∇×(∇×E⃗)=∇×−∂B⃗∂t=−∂∇×B⃗∂t.(1.19)
Now, we use the vector calculus identity expressed in equation (1.20),
∇×(∇×A⃗)=∇(∇·A⃗)−∇2A⃗.(1.20)
Using the identify of equation (1.20), in equation (1.19),
∇×(∇×E⃗)=∇(∇·E⃗)−∇2E⃗=−∂∇×B⃗∂t.(1.21)
The last term of equation (1.21) can be replaced using Ampère’s law, equation (1.17),
∇×B⃗=μ0J⃗+ε0∂E⃗∂t,(1.17)
thus, replacing equation (1.17) in equation (1.21),
∇×(∇×E⃗)=∇(∇·E⃗)−∇2E⃗=−∂μ0J⃗+ε0∂E⃗∂t∂t.(1.22)
Now, with the Gauss law we can reformulate equation