The ratio of the amplitudes of the electric field (E0) and magnetic field (B0) in an electromagnetic wave is equal to the speed of light (c). This can be derived from Maxwell's equations and the properties of electromagnetic waves.


Maxwell's equations describe the behavior of electric and magnetic fields in space. They consist of four equations:

1. Gauss's Law for Electric Fields: ∇ · E = ρ/ε0
2. Gauss's Law for Magnetic Fields: ∇ · B = 0
3. Faraday's Law of Electromagnetic Induction: ∇ × E = -∂B/∂t
4. Ampere's Law with Maxwell's Addition: ∇ × B = μ0J + μ0ε0∂E/∂t

These equations relate the electric and magnetic fields to each other and to the sources of electric charge (ρ) and electric current (J). From these equations, we can derive the wave equation for electromagnetic waves.


By combining Faraday's Law and Ampere's Law, we can derive the wave equation for electromagnetic waves:

∇^2E - μ0ε0∂^2E/∂t^2 = 0

Similarly, for the magnetic field:

∇^2B - μ0ε0∂^2B/∂t^2 = 0

The wave equation shows that the electric and magnetic fields in an electromagnetic wave are related to each other through the permeability of free space (μ0) and the permittivity of free space (ε0).


The wave equation solutions for an electromagnetic wave traveling in a vacuum have the form:

E(x, t) = E0sin(kx - ωt)
B(x, t) = B0sin(kx - ωt)

Where E0 is the amplitude of the electric field, B0 is the amplitude of the magnetic field, k is the wave number, x is the position, t is the time, and ω is the angular frequency.

Comparing the two equations, we can see that the ratio of the electric field amplitude (E0) to the magnetic field amplitude (B0) is constant:

E0/B0 = c

Where c is the speed of light in a vacuum, approximately equal to 3 × 10^8 m/s.

Therefore, the correct answer is option 'D': (E0/ B0) = c.