Deriving the Speed of Electromagnetic Waves From Maxwell's Equation in Vacuum and Non-Conducting Mediums

Key Takeaways

• Permittivity and permeability are two parameters that influence the speed of an electromagnetic wave through a medium.

• The net magnetic field lines through any closed surface are zero. This is proof of the non-existence of magnetic monopoles. 

• According to Ampere’s Law, the integral of the magnetic field around a loop is equal to the current running through the loop, multiplied by the permeability, µ.

The speed of electromagnetic waves differs in vacuum and non-conducting mediums

The speed of electromagnetic waves differs in vacuum and non-conducting mediums. Regardless of the medium, permittivity and permeability are the two parameters that influence the speed of electromagnetic waves as they travel through a medium. Luckily, we can mathematically explain this speed using Maxwell’s equations in vacuum and non-conducting mediums. 

Explaining Maxwell’s Equation

Let’s try to envision a positive charge enclosed in a closed surface. The field lines projecting out of the closed surface are equal to the net charge contained inside it, divided by the permittivity, 𝜀. This forms the first Maxwell’s equation, which can be mathematically given as:

Now, let’s consider a similar case on a magnet. Magnetic field lines emerge from the north pole and reach the south pole. Therefore, the net magnetic field lines through any closed surface are zero. This is proof of the non-existence of magnetic monopoles. Mathematically, Maxwell’s second equation can be given as:

To derive the third and fourth Maxwell’s equations, consider the equation below. We know that according to Faraday’s laws, the voltage around the loop is equal to the rate of change of flux through it.

Equation (4), above, provides Maxwell’s third equation.

According to Ampere’s Law, the integral of the magnetic field around a loop is equal to the current running through the loop, multiplied by the permeability, µ. Maxwell added a new term called displacement current into Ampere’s law, forming Maxwell’s fourth equation (given below).

Calculating the Speed of Electromagnetic Waves With Maxwell's Equation in Vacuum and Non-Conducting Mediums

In a vacuum, the current density, J, is equal to 0. Differentiating Maxwell’s third equation on both sides gives us:

Substituting equation 4 into the above equation leaves us with:

Compare equation (11) with the wave equation given below:

We can conclude that the speed of the electromagnetic wave in vacuum is the reciprocal of the square root of permeability of vacuum multiplied by the permittivity of vacuum.

In the case of a non-conducting medium with relative permeability 𝜇r  and permittivity 𝜀r , the speed of the electromagnetic wave through it is given by the following equation:

As we’ve demonstrated, the speed of an electromagnetic wave can be derived using Maxwell’s equation in vacuum and non-conducting mediums. When you are working with electromagnetic waves, Cadence’s software can help you model wave characteristics through any medium. 

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