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The Lossless Medium for Electromagnetic Waves

Key Takeaways

  • A lossless medium is a medium with zero conductivity and finite permeability and permittivity. 

  • When an electromagnetic wave propagates through a lossless medium, the amplitude of its electric field or magnetic field remains constant throughout the propagation.

  • The properties of the lossless medium affect the speed of propagation, and it is reduced by a factor of 1/ ( √𝝁r𝜺r) compared to the speed of the electromagnetic waves in the vacuum. 

Electromagnetic waves

Electromagnetic waves are defined using wave equations that describe the electric and magnetic fields of a wave. The solution of the wave equation describes the characteristics of electromagnetic waves—the velocity of propagation, frequency, attenuation, wavelength, and the direction of propagation. 

The characteristics of an electromagnetic wave vary with parameters such as the conductivity, permeability, and permittivity of the medium. A medium can be considered a lossless medium for electromagnetic waves when its conductivity is equal to zero, with specific permeability and permittivity values. In this article, we will discuss electromagnetic waves in a lossless medium.

The Lossless Medium for Electromagnetic Waves

A lossless medium for electromagnetic waves is a medium with zero conductivity (𝞂) and finite permeability (𝝁) and permittivity(𝜺). It can be described using the equation below:

 lossless medium for electromagnetic waves

Consider a plane wave propagating in the y-direction, with electric and magnetic fields mutually perpendicular to each other and to the wave propagation. A set of wave equations are used to define the electromagnetic waves:

plane wave propagating in the y-direction

The general solution to the linear, homogenous second order differential equations can be given in equations 4 and 5.

 general solution to the linear, homogenous second order differential equations

From the above equations, we can conclude that other than the field amplitudes, E1, E2, H1, and H2, the electromagnetic wave characteristics of the fields are identical. Characteristics of the electromagnetic waves defined by the general field solutions can be determined by investigating the corresponding instantaneous electric or magnetic fields. Taking this into consideration, the time-domain solution of the wave equation in terms of the electric field can be represented by equation 6.

 time-domain solution of the wave equation

In the equation, 𝛂 is the attenuation constant, 𝛃 is the phase constant, and 𝛚 is the angular velocity of propagation.

The Attenuation and Phase Constants

The attenuation constant is the measure of the attenuation of the amplitude of the fields in electromagnetic wave propagation. It can be described by equation 7. As the wave moves forward, its phase may undergo variations, and the factor that indicates this variation is called the phase constant and is shown in equation 8.

phase constant

By substituting 𝞂=0 in the above equations, we obtain the following:

𝛂=0                  (9)

𝛃=𝛚√𝝁𝜺               (10)

Equation 9 is the mathematical representation of no attenuation in the lossless medium. When an electromagnetic wave propagates through a lossless medium, the amplitude of its electric field or magnetic field remains constant throughout the propagation. There is no reduction to the amplitudes of the wave in the course of propagation through the lossless medium. In a lossless medium, electromagnetic waves can be called non-decaying waves. 

The phase of the electromagnetic wave changes in a lossless medium, and is influenced by the velocity of wave propagation and the characteristics of the medium. Equation 10 shows that as the value of the relative permeability and relative permittivity of the medium increases, the phase constant also increases. The angular velocity of the wave matters when coming to the phase variations of the electromagnetic wave.

 Now, let’s discuss the effect of a lossless medium on the velocity of wave propagation.

The Velocity of Wave Propagation

Velocity of Wave Propagation

The velocity, u, of the electromagnetic wave is given by equation 11. Since we already know what the value of 𝛃 is, we can rewrite the velocity as equation 12. 

rewrite the velocity as a new equation

From the equation, we can see that the properties of the lossless medium affect the speed of propagation, and it is reduced by a factor of 1/(√𝝁r𝜺r) compared to the speed of the electromagnetic waves in the vacuum.

The free space or vacuum can be approximated to a lossless medium for electromagnetic waves. Cadence’s software can help you understand the various parameters associated with electromagnetic wave propagation through any medium. 

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