Maxwell’s equations predict that the speed of light is constant

The aim of this proof is to transform the Maxwell’s equations into an equation that describes electromagnetic waves (the one-dimensional wave equation):

∇2E⃗ =1v2∂2E⃗ ∂t2

This equation allow us to calculate the speed of an electric wave (v) which is equal to the speed of light. With a mathematical development from Maxwell’s equations, we will see that the speed of light only depends on constants. This idea of a constant speed of light is historically very important for modern physics since it gave Albert Einstein the idea to develop the theory of relativity.

Prerequisite information

  • Be able to use these four vector operators:
  1. Curl: ×
  2. Divergence: 
  3. Gradient: 
  4. Laplacian: 2
  • You need to know the basics about Maxwell’s equations.
  • You also need to know the one-dimensional wave equation mentioned above.

The proof

Here are the four Maxwell’s equations for an electric field E⃗  and a magnetic field B⃗  in a linear medium:

The proof

We will calculate the speed of light in the vacuum, this is why we replace μ (permeability) and ε (permittivity) by μ0 (vacuum permeability ≈ 1.256×10−6) and ε0 (vacuum permittivity ≈ 8.854× 10−12). In order to get the result we must consider that, because we are in the vacuum, there is no charge. This is why the current density (j⃗ ) and the charge density (ρ) are equal to 0. After making those modifications, we get:

The proof 2

In order to get the wave equation, we first have to isolate the electric field E⃗ 2E⃗ =1v22E⃗ t2.. To do this, we have to take the curl of the equation (2) and differentiate the equation (4) with respect to time.

The proof 3

We can see that in the right-hand side of (5), there is a derivation with respect to time and then a derivation with ×. In the left-hand side of (6) it is the opposite case. Because the derivative is a commutative operation, we have can equalize -(5) and (6):

The proof 4

We substitute the right hand-side of the equation (5) by right-hand side of the equation (6):

The proof 7.

The electric field E⃗  is now isolated. However, there is no Laplacian 2, this is why it does not look like the wave equation of the electric field The laplacian in the wave equation : 2E⃗ =1v22E⃗ t2.. In order to get the Laplacian in the equation (7), we have to prove this equality:

×(×E⃗ )=(E⃗ )2E⃗ 

By developing both sides of the equation with the vector notation, we get:

By developing both sides of the equation with the vector notation, we get:

By using the properties of the derivative, we prove that the equality (8) is correct:

By using the properties of the derivative, we prove that the equality (8) is correct:

Now, we know that both (7) and (8) are correct. We combine them in one single equation:ε0μ02E⃗ t2=(E⃗ )2E⃗ 

As stated at the beginning of the proof, E⃗ =0 because we calculate the speed of light in vacuum (see equation (1)). This is why we get:

Now, we know that both (7) and (8) are correct. We combine them in one single equation:  −ε0μ0∂2E⃗ ∂t2=∇(∇⋅E⃗ )−∇2E⃗   As stated at the beginning of the proof, ∇⋅E⃗ =0 because we calculate the speed of light in vacuum (see equation (1)). This is why we get:

If we compare the equation (9) with the one-dimensional wave equation  2E⃗ =1v22E⃗ t, we can deduce that:

If we compare the equation (9) with the one-dimensional wave equation  ∇2E⃗ =1v2∂2E⃗ ∂t2 , we can deduce that:

Where ϵ0 Vacuum permittivity and μ0 Vacuum permeability are both constant. This is why the speed of light is constant in the vacuum. It is possible to extend this principle to other references, because Maxwell’s equations are Lorentz invariant #

Note

Note that there are two way to interpret this result:

  1. The speed of light was constant with respect to the aether. This theory turned out to be false. Read more about the Luminiferous aether.
  2. The theory of Relativity

Binomial theorem

The binomial theorem states that x,y and n,k, such as kn, we have:

Binomial theorem

Inductive Proof

Prerequisite information

  • Have a knowledge of binomial coefficient.
  • Have a knowledge of the principle of induction.
  • Know that: x0=1.

The proof

In order to show that the binomial theorem is true for n. We will first prove it is true for n=0, then for n by induction.

Proof for n=0

We have:

(x+y)0=1

and

Proof for n=0
We then deduce that the binomial theorem is true for n=0.

Inductive step

We know that for n=0, we have:

Inductive step

For n+1, we have:

For n+1, we have:

This is why we have:

This is why we have

Because we know that the binomial theorem is true for n=0 and for n=n+1, we deduce by induction that it is true for n.

Fundamental theorem of calculus

The fundamental theorem of calculus is a like between the concept of the derivative of a function and the concept of integral. There are two parts to the theorem.

First part

This part is also known as the first fundamental theorem of calculus.

The theorem

Let f be a continuous real-valued function which is continuous on the closed interval [a,b]. Let F be the function defined, for all x in [a,b], by the following equation:

The theorem

Then, F is a primitive of f on [a,b].

Prerequisite information

  • Have a knowledge of the definition of the primitive.
  • Know the definition of the derivative.
  • Know the mean value theorem.

The proof

We want to find the primitive of F, denoted F. This is why we use the defintion of the derivative to get started:

The proof

We know that f is continuous on [a,b], this is why it is continuous on [x,x+h[x;x+h][a;b]. From the mean value theorem, we know that there exist a point c[x,x+h] such that:

The proof 1

Then, we deduce from the results (1) and (2) that:

The proof 2

When h0cx, this is why we have:

The proof 3

Second part

This part is also known as the second fundamental theorem of calculus. Sometimes it is referred to as the evaluation theorem (in context of integral calculus).

The theorem

Let F be a primitive of f on [a,b]. If f is continuous on the interval [a,b], then:

The theorem

Prerequisite information

  • Know the first part of the fundamental theorem of calculus.

The proof

From the first part of the fundamental theorem of calculus, we have:

The proof

Then, we deduce that:

Then, we deduce that