Speed of Light: UNIZOR.COM - Physics4Teens - Waves - Field Waves
Channel: Zor Shekhtman on UNIZOR Education
Date: 22/10/2022

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

Speed of Light

Based on Maxwell equations, we can derive separate differential equations for vectors of electrical (E) and magnetic (B) components of a electromagnetic field and the speed of its propagation in a simple case of vacuum as a medium and no additional sources of electric charges, moving or stationary, present.

As we know, oscillations of electromagnetic field are transverse, vectors E and B are perpendicular to each other and both are perpendicular to the direction of waves propagation.

Assume that the wave propagation is along the Z-axis of some system of coordinates, electric component E is changing along the X-axis and magnetic component B is changing along the Y-axis.

For this type of the electromagnetic field we can state
E = {Ex,Ey,Ez}
with Ey=0 and Ez=0,
while Ex(t,z) being a function of time t and coordinate along the Z-axis z.
Analogously,
B = {Bx,By,Bz}
with Bx=0 and Bz=0,
while By(t,z) being a function of time t and coordinate along the Z-axis z.

Using the above format of vectors of electrical (E) and magnetic (B) components of the electromagnetic field, we can analyze the form of the Maxwell equations for them.

The simplification of the third and fourth Maxwell equations for this type of an electromagnetic field leads us to the following two equations
∂²Ex/∂z² = −∂²By/∂t∂z
−∂²By/∂z∂t = μ·ε·∂²Ex/∂t²
From these two equations we can construct one for Ex:
∂²Ex/∂z² = μ·ε·∂²Ex/∂t²

In vacuum magnetic permeability μ is a known constant, approximately equal to
μ0=1.25663706212·10^(−6) Henry/meter
and an electric permittivity ε is also a known constant, approximately equal to
ε0=8.85418781281·10^(−12) Farad/meter.

One of the solutions to it, considered in the lecture "Wave Equation 2" of this chapter of the course on UNIZOR.COM is the wave function
E(t,z) = A·sin(ω·(t−z/v))
where ω is the frequency of oscillations and v is the speed of wave propagation.

If 1/v²=μ0·ε0, the wave equation for E(t,z)=A·sin(ω·(t-z/v)) is satisfied.

The last statement is extremely important, since it defines the speed of wave propagation in terms of magnetic permeability and electric permittivity of the medium.

Using the constants of magnetic permeability and electric permittivity of vacuum listed above, the speed of propagation of electromagnetic oscillations for vacuum can be calculated as follows:

1/v² = μ0·ε0 = 11.1265005604·10^(−18)
v² = 1/(μ0·ε0) = 0.0898755178·10^(18)
v = √(0.0898755178)·10^(9) = 299,792,458 m/sec

The speed of light in vacuum is usually denoted as c, so
c = 299,792,458 m/sec