Question

Show that `1/sqrt(ε_0µ_0)` gives the velocity of an electromagnetic wave in free space.

Answer 1

Maxwell obtained the wave equation for electric and magnetic fields in free space as:

`∇^2E = mu_0ε_0(∂^2E)/(∂t^2)`

This equation is comparable to the general wave equation:

`∇^2ψ = 1/v^2(∂^2ψ)/(∂t^2)`

On comparing the two equations, we get:

`1/v^2 = mu_0ε_0`

v = `1/sqrt(mu_0ε_0)`

Substituting the values:

µ₀ = 4π × 10⁻⁷ T m A⁻¹

ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻²

v = `1/(sqrt((4pi xx 10^-7)(8.85 xx 10^-12))`

= 3 × 10⁸

Hence, the speed of electromagnetic waves in free space is:

c = `1/sqrt(mu_0ε_0)`

This result shows that electromagnetic waves travel at the speed of light, thereby confirming that light is electromagnetic in nature.