Group A Intro

Electromagnetic fields

The most common description of the electromagnetic field uses two three-dimensional vector fields called the electric field and the magnetic field . These two vector fields serve as mediators in interactions among charged material objects. This formulation stands in contrast to the previous notion of “action at a distance,” predating field theory, where forces seemed to act directly between objects without intermediary fields.

We identify the two fields:

In materials that exhibit both dielectric and magnetic properties, such as ferromagnetic materials, the behavior of electromagnetic fields change. This is due to the interaction of the electromagnetic field with the intrinsic properties of the material.

When an electromagnetic field propagates through such a material, it induces polarization and magnetization within the material, which contribute to the overall response of the medium to the electromagnetic field. This response is characterized by the introduction of additional fields.

The electric displacement field () and the magnetic field intensity ():

Electric displacement field ()

The electric displacement field has two components:

• is the component that takes into account the electric field in vacuum.
• is the polarization vector and represents the tendency of dipoles in a material to align, in the case of permanent dipoles, or to be temporarily created, in the case of induced dipoles, by means of an external electric field. is defined as the electric moment per unit volume. The polarization term contributes to the electric displacement field by introducing an additional displacement of charge within the material.

Magnetic field intensity ()

The constant is the magnetic permeability of the vacuum and is the magnetization of the medium, which is defined as the magnetic moment per unit volume. In a vacuum,  and  are proportional to each other. Inside a material they are different due to .

Linear, homogenous, isotropic medium

We start defining the electric susceptibility that is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field.

Info

Homogeneous means that the properties of the material are constant at any point (within the material). i.e. does not depend on position.

Isotropic means that a material has the same properties in all directions. In other words, the response of the material to an electric field is the same regardless of the direction of the field. i.e. is a scalar (otherwise is a tensor)

Linear means that in this materials the response is proportional to the excitation. Response of the material is directly proportional to the applied field. i.e. is independet on

Instantaneous means that the response of the medium to the electric field excitation follows instantaneously the excitation

In a linear, homogenous, isotropic medium the microscopic dipoles align along the direction of the applied electric field, so that we can write:

where

• is the electric permittivity of the vacuum
• is the electric susceptibility
• is the relative dielectric permittivity of the medium

We can also find a simple expression for the magnetization of the medium . This formula, together with the definition of magnetic field, gives us

where is the relative magnetic permeability of the medium.

We will always assume unless otherwise specified

This is due to the fact that the magnetic dipoles are too slow to respond. #todo not too slow but weak, ask to prof?

Maxwell’s equations

where is the free charge density and is the free current density.

1. is the Gauss’ law of electrostatics
2. is the Gauss’ law for magnetostatics (with the assumption of no magnetic monopoles)
3. is the Faraday and Lenz law of magnetic induction
4. is the Ampere and Maxwell law

Quantization of EM radiation cannot be described by the MEs

Electromagnetic waves

The Maxwell’s equations are four partial differential equation, and is possible to find a wave-like solution for the and considering the special case in which we are inside a linear, isotropic, homogeneous, non magnetic material and (no free charges).

Under these conditions we see that and that so we can substitute these into the 4th MEs and get:

taking the curl of both members of and using the previous relations:

now using the vector identity:

the simplification of the divergence term is due to the that in these special condition is so we obtain this single PDE (Partial derivative eq) for the electric field:

and analogously for the magnetic field:

Inside the dielectric material the propagation velocity is given by:

where is the refractive index of the medium, which allows us to relate the optical properties of a medium to its dielectric properties.

The refractive index can be seen as the factor by which the speed of the radiation is reduced with respect to its vacuum value, for these reason the refractive index of the vacuum is 1

The D’Alembert equation

The D’Alembert equation is a partial differential equation that is commonly used to describe the solution of the one-dimensional wave equation. It can be used to describe the MEs and is defined as:

where:

• its the speed at which the wave propagates.
• is the displacement of the wave at position and time

We can rewrite equations (1) and (2) using a 3D formulation of the D’Alembert equation:

these helps in finding solutions of the MEs.

Plane waves

Plane waves are a particular class of waves, that is represented by those waves where and only depends on a direction, say , and on time.

From the MEs directly follows that and oscillate perpendicularly to the direction of the wave’s advance so they are transverse waves ().

Each non null component of the electric and magnetic field must solve the 1D D’Alembert equation. In the hypothesis that these components are respectively directed along the −axis for the electric field and the −axis for the magnetic field, we get:

the general solution of which is the sum of a progressive and regressive wave with the form:

where:

• represents a forward-traveling wave
• represents a backward-traveling wave.

Notice that to be a proper solution of the D’Alembert equation, a function must depend only on .

and are not independent:

\frac{\mathcal{E}}{B}=v

Why did we spend valuable time to delve into the concept of plane waves?

Well, even if the plane wave is a mathematical abstraction, as there is no physical source infinitely extended in space that can produce it, several more complex wave propagation can be approximated by plane waves under certain conditions. For instance plane waves are often used to describe the propagation of light in homogeneous media over large distances.

Harmonic plane waves

A particular case of plane waves traveling along the −axis is the harmonic (monochromatic) plane wave whose wave function depends periodically on the argument ().

In general a harmonic plane wave is represented by a solution of the 3D wave equation where and depend on the position vector and the time as harmonic functions:

They reduce the study of a 3-component vector field to a 1-component scalar function. This is because these fields are harmonic functions, meaning they repeat themselves over time and space with a constant frequency and wavelength.

We can describe the electric and magnetic field solution by complex numbers, in order to simplify the calculation and we obtain:

Notice that in the first formula above both the complex number and its real part are called .

Of course physically measurable quantities are obtained by taking the real part of the complex wave

In conclusion, we found that and are mutually orthogonal and orthogonal to the propagation direction.

Energy of a plane wave and Poynting vector

The Poynting vector, named after John Henry Poynting, is used in order to demonstrate the energy flux density of an EM field. Per definition, the Poynting vector is the result of the vector product of the field’s electric and magnetic components:

the Poynting vector provides information about the direction of propagation of the EM field and information about the direction of energy transport.

Both the energy flux and wave propagation velocity share the same direction in the case of a harmonic plane wave, it implies that the energy carried by the wave is moving in the same direction as the wave itself.

Now, the average intensity or intensity is defined as the average of the intensity over one period of the wave. Intensity, in the context of waves, typically refers to the power per unit area carried by the wave. It’s a measure of how much energy is transmitted through a given area per unit time. We can calculate the average intensity using the wave impedance that in case of the ideal dielectric (conductivity is zero) is

so we can write:

Polarization of EMW

The direction of the electric field of a plane wave is called polarization.

This polarization is completely different from the macroscopic polarization of the material due to the presence of an electric field; they should not be confused.