Using Maxwell equations, show that the speed of light, c is 2.99792 × 108 m s-1

Sure, let's derive the speed of light using Maxwell's equations.

Maxwell's equations are a set of four differential equations that describe how electric and magnetic fields interact. The equations are:

1. Gauss's law for electricity: ∇ • E = ρ/ε₀
2. Gauss's law for magnetism: ∇ • B = 0
3. Faraday's law of induction: ∇ x E = - ∂B/∂t
4. Ampère's law with Maxwell's addition: ∇ x B = μ₀J + μ₀ε₀ ∂E/∂t

We can derive the speed of light from the last two equations.

First, take the curl of both sides of Faraday's law of induction:

∇ x (∇ x E) = ∇ x (- ∂B/∂t)

Using the vector identity ∇ x (∇ x A) = ∇(∇ • A) - ∇²A, we get:

∇(∇ • E) - ∇²E = - ∂/∂t (∇ x B)

Substitute Gauss's law for electricity (∇ • E = ρ/ε₀) and Ampère's law with Maxwell's addition (∇ x B = μ₀J + μ₀ε₀ ∂E/∂t) into the equation:

∇(ρ/ε₀) - ∇²E = - ∂/∂t (μ₀J + μ₀ε₀ ∂E/∂t)

Assuming free space where there are no charges (ρ = 0) and no currents (J = 0), the equation simplifies to:

• ∇²E = - μ₀ε₀ ∂²E/∂t²

This is a wave equation for the electric field E with wave speed c = 1/√(μ₀ε₀).

The values of the permeability of free space μ₀ and the permittivity of free space ε₀ are known and measured in SI units as μ₀ = 4π x 10^-7 T m/A and ε₀ = 8.85418782 x 10^-12 C²/N m².

Substituting these values in, we get:

c = 1/√(μ₀ε₀) = 1/√[(4π x 10^-7 T m/A)(8.85418782 x 10^-12 C²/N m²)] = 2.99792458 x 10^8 m/s

So, the speed of light c is approximately 2.99792 x 10^8 m/s, as required.