Question

c^2=1/epsilon zero (mew zero)
What's this formula and
How it's derived?

Answer 1

Explanation:

What is the formula

c = 1/√(ε₀μ₀)

This equation gives an expression for speed of electromagnetic waves in vacuum in terms of absolute permitivity(ε₀) and absolute permeability(μ₀) of vacuum. In general, speed of electromagnetic wave in any medium is given by

c = 1/√(εμ)

where ε and μ are permitivity and permeability of the medium.

The equation is a result of deriving a differential equation of electromagnetic waves from Maxwell's Equation.

Maxwell's Equations

Maxwell Equations for electromagnetism is a set of four equations relating electric field (E), magnetic field(B), electric charge density(ρ) and current density(j). The equations are

∇.E = ρ/ε₀

∇.B = 0

∇xE = - ∂B/∂t

∇xB = μ₀(j + ε₀∂E/∂t)

Here

∇ = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k

Deriving Equations of EM Waves

Let us consider free space where there is no charge or current. In this case, the equations reduce to

∇.E = 0 ..............................(1)

∇.B = 0 ..............................(2)

∇xE = - ∂B/∂t .....................(3)

∇xB = μ₀ε₀∂E/∂t ................(4)

Taking curl on both sides of (3) gives

∇(∇xE) = - ∇x(∂B/∂t)

∇(∇.E) - ∇²E = -∂(∇xB)/∂t

Using (1) and (4) we get

- ∇²E = -μ₀ε₀∂²E/∂t²

∇²E = μ₀ε₀∂²E/∂t²

where ∇² is the Laplace Operator.

In the similar fashion we can prove

∇²B = μ₀ε₀∂²B/∂t²

Both equations have the form of general equation of wave in 3- dimensional space with speed v

∇²Ψ = (1/v²)∂²Ψ/∂t²

Equating the co-efficients on the right side, we derive the speed of the electric and magnetic waves which is generally denoted by c.

1/c² = μ₀ε₀ = ε₀μ₀

c² = 1/ε₀μ₀

c = 1/√(ε₀μ₀)

Consequences

In the derivation process, no frame of reference was taken into consideration. It means that the speed of electromagnetic wave is independent of frame of reference. This thought baffled Einstein for many years. Later on, the fact that 'Speed of light is the same irrespective of the speed of observer' became an important postulate of Einstein's famous Special Theory of Relativity which revolutionized our views about the universe.