The 4 equations of electricity and magnetism yield waves that propagate through a vacuum. 2 electromagnetic waves with different amplitudes start at the same position and travel simultaneously.
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電場の振幅を2倍にすると、波の速さは何倍か？
比誘電率4の物質の中で、3 m進むのにかかる時間は真空の何倍か？
The equations explain the phenomenon
1The electric flux through a closed surface is calculated.∮ E · dS=Qε₀……
2The net outward electric flux from a small box is divided by its volume.∇ · E=ρε₀……
3The divergence of the magnetic field is obtained from the magnetic flux through a closed surface.∇ · B=0……
4Faraday's law of induction is expressed in terms of the curl of the electric field.∇ × E=−∂B∂t……
5The magnetic field produced by a steady electric current is expressed in terms of its curl.For a steady electric current:∇ × B=μ₀ J……
6Applying the steady-current equation during charging reveals a contradiction.∇ · (∇ × B)=0 ≠μ₀ ∇ · J……
7Adding the displacement current density extends the equation to time-dependent situations.∇ × B=μ₀ J+μ₀ ε₀ ∂E∂t……
8In a vacuum, the curl of the electric field equation is taken once more.∇ × (∇ × E)=−μ₀ ε₀ ∂²E∂t²……
9A vector identity yields the wave equation.∂²E∂t²=(1μ₀ε₀) ∇²E……
10Represent the amplitude by A and substitute the wave expression into the wave equation.Ey=A f(x − c t)：c is independent of A.……
11The effect of polarization is included in the permittivity ε to determine the wave speed.v=1√(ε μ₀)=c√εr……
Further study: numerical calculations on a grid and energy
12The equations are solved on a grid.∂Bz∂t=−∂Ey∂x，∂Ey∂t=−c² ∂Bz∂x……
13The energy flow in an electromagnetic wave is calculated.S=E × Bμ₀、|S|=ε₀ c |E|²(A traveling plane wave in a vacuum)……
What this chapter has shown
The laws of electricity and magnetism can be expressed as 4 equations involving divergence and curl. Adding displacement current resolves the inconsistency, and the electric and magnetic fields satisfy wave equations even in vacuum. Their propagation speed, 1/√(μ₀ε₀), equals the speed of light and is independent of amplitude.
A factor of 1.00Speed ratio when the amplitude is doubled
For a relative permeability of 1 and a relative permittivity of 4, the speed is c/2, and traveling 3 m takes 20.01 ns.
In everyday lifeIn the glass of an optical fiber, light travels at about 2/3 of its speed in vacuum. In the linear regime, its speed is independent of its intensity. In real glass, ε and μ also depend on the frequency of the light.
Detailed summary
The laws of electricity and magnetism can be expressed as 4 equations using divergence and curl. Ampère's law in its original form conflicts with conservation of electric charge; adding the displacement current resolves this conflict. The resulting equations predict waves in which electric and magnetic fields generate each other, even in a vacuum. Their speed, 1/√(μ₀ε₀), matches the speed of light. The amplitude cancels from both sides of the wave equation, so the wave speed is independent of amplitude. The amplitude determines the energy carried by the wave, which is proportional to the amplitude raised to the power 2. In matter, ε₀ is replaced by εrε₀, and the speed decreases to c/√εr.