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 (1.22), thus let’s recall the Gauss law,
∇·E⃗=ρε0,(1.9)
replacing equation (1.8) in equation (1.22),
∇ρε0−∇2E⃗=−μ0∂J⃗∂t−μ0ε0∂2E⃗∂t2.(1.23)
If we are working in a charge- and current-free region, ρ=0 and J⃗=0, then equation (1.23) becomes,
∇2E⃗=μ0ε0∂2E⃗∂t2.(1.24)
Equation (1.24) is called the wave equation. The wave equation is an important second-order linear partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves, and waves in water. It is important in various fields such as acoustics, electromagnetism, quantum mechanics, and fluid dynamics. The form represented in equation (1.24) is for an electric field E⃗. But if we apply ∇⃗× to Ampère’s law, then apply a similar procedure we can get,
∇2B⃗=μ0ε0∂2B⃗∂t2.(1.25)
Equation (1.25) is the wave equation for B⃗. The process to get equation (1.25) is left as an exercise to the reader.
1.9 The speed and propagation of light
The speed that a wave propagates is presented in the standard
