Electromagnetic waves

Wave in an ideal dielectric We will derive the wave equations for electric and magnetic field from Maxwell’s equations. It will be a proof that the electromagnetic field is a wave. Assumptions: Considered area is unlimited Considered medium is linear, homogenous and isotropic γ=0, it means a medium is lossless There are no currents and no charges in the considered area

We have a set of four differential equations with two unknown vector functions. In a general case we have to find 6 unknown scalar functions – components of searched vectors.

Assumptions: μ=const., ε=const., γ=0, ρ=0 Let’s calculate the rotation of both sides of both equations

From vector identity:

=0 =0

We have received two wave equations. Two vector functions E(P,t) i H(P,t) which satisfy this equations form electromagnetic wave. Attention: This means there might exist such solutions of wave equations which don’t satisfy Maxwell’s equations. Solutions E(P,t) and H(P,t) must satisfy condition coming from Maxwell’s equations.

Let’s use the x,y,z coordinates system: Vector Laplasian: Each of components satisfies scalar wave equation.

Then the wave equation is: Let’s consider only one of the field components, but received conclusions will be general. Assumption: The field changes occur in one direction only, let’s assume it is z direction. It means that all derivatives with respect of x and y equal zero. Plane wave Then the wave equation is:

The solution of this equation has a form of: When the shape of function f depends of the wave generator. a velocity of the wave propagation

There are equations of uniform motion. Let’s assume f1=const. and f2=const. What does result from this assumption? There are equations of uniform motion. Points in which f1 has a constant value move in positive direction of the z axis. f1 – a wave moving ahead called ORIGINAL WAVE Points in which f2 has a constant value move in negative direction of the z axis. f2 – a wave moving back called RETURNED WAVE

What is the result of the fact that the changes occur in z direction only (in the direction of wave propagation)?

Ex,Ey,Hx,Hy are functions of After integration we receive Ez=const. i Hz=const. Ez=0 Hz=0 The field is produced by the changes. Constancy is contradictory to the assumption of field existence. CONCLUSION: Considered wave doesn’t have components in direction of propagation. Ex,Ey,Hx,Hy are functions of z-vt, time and variable z.

Ex=f1(z-vt), Ey=f2(z-vt), Hx=f3 (z-vt), Hy=f4 (z-vt) Def.: A wave is called transverse when wave vector doesn’t have components in direction of wave propagation. This wave is marked TEM – transverse electromagnetic Ex,Ey,Hx,Hy - are solutions of independent wave equations. Moreover, they have to satisfy Maxwell’s equations. Let’s mark: Ex=f1(z-vt), Ey=f2(z-vt), Hx=f3 (z-vt), Hy=f4 (z-vt)

f1 f4 f3 f2 integration Constants of integration are omitted, because they don’t satisfy the assumption of wave existence.

After substitution f1, f2, f3, f4 : x y z x y z Hy v Hx v Ex -Ey

- The wave admitance of medium - The wave impedance of the medium - The wave admitance of medium The impedance is the ratio of the values of mutually perpendicular components of electric and magnetic wave vectors.

The scalar product: Scalar product =0 when vectors are perpendicular.

Twierdzenie Poyntinga Poynting’s theorem Poynting’s theorem determines the law of power conservation in area V in which electric and magnetic fields exist. Assumptions: A wave propagates in homogenous medium ε=const μ=const γ=const

Maxwell’s equations: Let’s subtract both sides of both equations

From vector identity: we receive We integrate this equation over the volume V. The area V is bounded by the surface S.

Using Gauss’s theorem: where S – the surface bounding the area V and vector dS is directed along the external normal.

Let’s assume that any kind of energy is transformed into electric energy in a part of the volume V. An electric energy sources are situated in the considered area.

Let’s rearrange the terms:

After substitutions: Volume density of the generated power Power generated in area V Volume density of the power connected with the transformation of electric energy into the heat Power connected with the transformation of electric energy into the heat in area V.

Volume density of electric field energy Volume density of magnetic field energy Electromagnetic field energy in area V A derivative of energy is the power of electromagnetic field in area V.

Poynting’s vector direction determines the direction It has a physical sense of surface density of the power. Poynting’s vector flux. Physically it represents the power flux through the surface S. Poynting’s vector direction determines the direction of the power flow.

V The flux of the Poynting vector through the surface S is called the power flux or the power radiated through this surface. The power radiated through the bound of the area V. V Input power Output

Poynting’s theorem: The power produced in any area V is equal to the sum of: the power transformed into heat the power gathered in electromagnetic field in this area the power radiated through the bound of V This theorem expresses the law of energy conservation in electromagnetic field.

Let’s write this law as a power balance: POWER RADIATED POWER GATHERED AND LOST POWER GENERATED

electromagnetic field Harmonic electromagnetic field

COMPLEX VECTORS Time vector W(t) has sinusoidal components: Def. The field is harmonic when vectors describing this field are sinusoidally changing in time. Time vector W(t) has sinusoidal components:

Def. Complex vector has complex components which are complex values of the time vector components. Relations between components of the time vector and complex vector.

The measure of the complex vector is its NORM Relation between the time vector and complex vector: The measure of the complex vector is its NORM Def. The norm of complex vector is expressed as:

MAXWELL’s EQUATIONS IN COMPLEX FORM

- complex vectors where under assumptoion:

Maxwell’s equations in complex form: and

Considered area is: linear isotropic uniform There are no charges in considered area.

Г - complex coefficient – propagation constant Complex equivalents of wave equations They are called the Helmholtz wave equations Г - complex coefficient – propagation constant

There are complex equations for components

where The solution of Helmholtz’s equation will be analogical for each component where -propagation constant damping constant phase constant

Physical interpretation of and Let’s consider the case Assumption:

Plane wave E2 E1 Assumptions: Plane wave equations in time and in complex forms Assumptions: Let’s consider one of the components, eg. Ex E1 E2

E1(t) E2(t)

A. Original wave B. Returned wave Wave length: An equation of the wave motion: A. Original wave B. Returned wave Wave length:

Wave amplitude Wave amplitude decreases with A. Original wave B. Returned wave Wave amplitude decreases with z axis, it means that amplitude is damped in direction of wave propagation Wave amplitude decreases with –z axis, it means that amplitude is damped in direction of wave propagation (-z)

From Maxwell’s equation:

H1 H2

Wave impedance of the medium in which the wave propagates There is no returned wave, when the medium is unlimited

Plane wave in ideal dielectric

! Assumption: there is no returned wave The wave is undamped Ex and Hy are in phase because Zf is real number

Energy divides into halves for electric and magnetic fields. Let’s compare the density of energy in electric and magnetic fields: Energy divides into halves for electric and magnetic fields.

Plane wave in ideal conductor

Damping constant and phase constant are equal in ideal conductor.

Argument of wave impedance equals to difference of initial phases of electric and magnetic fields.

Electric field leads magnetic field by 45 A damping is very large in conductor. The wave amplitude is almost entirely damped on a way equal one wave length. times After passing one wave length the wave in conductor fades.

Let’s compare energy densities in electric and magnetic fields:

Because the average value for the period where when In ideal conductor whole energy is included in magnetic field.

Polarisation of harmonic wave

x y z Hy Ex v Hx -Ey

Now let’s consider relations between Ex z Ey and Hx z Hy

maximum values initial phases of components Ex and Ey The end of the vector E draws the line which shape depends on maximum values initial phases of components Ex and Ey

( ) Eliptic polarisation ° + - = 90 2 , 10 sin 7 z t E y x j z=const. Eliptic polarisation Components have different amplitudes and different phases

( ) z=const. Eliptic polarisation in the axis x and y directions ° + - = 90 2 , 10 sin 7 z t E y x z=const. r Eliptic polarisation in the axis x and y directions Amplitudes of components are different and phases differ by 90

Amplitudes of components are equal and phases differ by 90° Circular polarisation Amplitudes of components are equal and phases differ by 90°

Linear polarisation Amplitudes are different, but the phases are the same.