Main concepts. When a plane EMW propagates from the source (which is located at point x0=0) along the positive direction of x-axis

; (23)

e₀ and m₀ – electric and magnetic constants correspondingly;

e and m – relational electric permittivity and magnetic permeability of medium (as a rule the transparent medium is non-magnetic m=1).

In one EMW the volume density of energy of electric field w _C is equal tovolume density of energy of magnetic field w _L:

. (24)

Instantaneous flux density of energy of EMW (Pointing’s vector)

. (25)

The average value of Pointing’s vector defines the wave intensity:

I=P _AVE = E_m× H_m / 2. (26)

EXAMPLE OF PROBLEM SOLUTION

Example 4. In the homogeneous isotropic non-magnetic medium with the dielectric permittivity of ε = 9 along the х -axis propagates plane EMW from wave source which is located at point x ₀ = 0. The change of intensity of magnetic field is described by equation H_z (x, t)= H_m× cos(w t – kx –p / 2), when amplitude of magnetic field intensity in a wave 0, 02 A/m. The oscillation period is1 m s.

1) To rebuild the equations of change of electric field intensity and magnetic field intensity with numerical coefficients.

2) To draw the graph of wave at the moment of time of t ₁=1, 5 T.

3) To define the Pointing’s vector at the moment of time of t ₁=1, 5 T in the point with coordinate x ₁ = 1, 25l and plot it on the graph.

4) To define the wave intensity.

Input data: Нm = 0, 02 A/m; Т =1 m s = 10 –6 s; ε = 9; μ = 1; Hz (x, t)= Hm× cos(w t – kx –p / 2)	Figure 3.6 –Graph of wave at t 1=1, 5 T.
Find: Нz (x, t), Еy (x, t) –? Graph, , I –?

Solution:

1) Equations of given EMW have a general view:

E_y (x, t)= E_m× cos(w t – kx +j₀); H_z (x, t)= H_m× cos(w t – kx +j₀),

where E_m and H_m – amplitudes of intensities of electric and magnetic fields of the wave correspondently, x – coordinate of a point of space; t – time of propagation of a wave; j₀= –p / 2 – initial phase of the wave source.

For the rebuilding this equations with numerical coefficients, it’s necessary to define the cyclic frequency ω and wave number k, which are defined by equations:

; (4.1)

. (4.2)

Period Т is given in the problem statement, the wave length λ is a distance which is transited by a wave for a period:

, (4.3)

where u – phase velocity of propagation of EMW. In non-magnetic medium with permeability μ =1 and dielectric permittivity ε the velocity u of propagation of EMW is defined with the formula:

, (4.4)

where с= 3× 10⁸ m/s – the speed of light.

Let’s substitute in the formula (4.2) the expression of λ from the formula (4.3) and u from the formula (4.4):

. (4.5)

From the equality of volume energy density of electric and magnetic field

,

we obtain the relation between the amplitudes of electric and magnetic intensities:

. (4.6)

The right part of the formula (4.1) gives the unit of measurement of cyclic frequency [ rad/s ]; let’s check whether the right part of the formula (4.5) gives us the unit of wave number [ rad/m ], and right part of the formula (4.6) – the unit of intensity of electric field [ V/m ].

;

Let’s make the calculations and write down the equation Е and Н with numerical coefficients

; ;

;

Then finaly equations of EMW: E_y (x, t)=2, 5 × cos(2× 10⁶p× t –0, 02p× x –p / 2) V /m;

H_z (x, t)=0, 02 × cos(2× 10⁶p× t –0, 02p× x –p / 2) A /m.

2) Let’s draw the graph of wave at the moment of time of t ₁=1, 5 T.

At this time the source will have a phase, equal to

F(x =0, t ₁)= (2× 10⁶p× 1, 5× 10 ^–6 – 0, 02p× 0– p / 2)= (3p – 0– p / 2)= p / 2,

then intensities of electric and magnetic fields in a source will have a zero values, as a cos(p / 2)=0 (Fig. 3.6, point x =0).

Through distance, equal to wave length

l= 2p / k or l= 2p / 0, 02p=100 m

this value will repeat, as a cos(p / 2– k l)=0 (see Fig. 3.6, point x =l).

During this time the wave will transit distance equal to position of a wave front:

,

then in position of wave front intensities of electric and magnetic fields in a source will have a values same as source at t =0, that is zero, as a cos(p / 2– k× 1, 5l)=0 (see Fig. 3.6, point x = x _WF).

3) Let’s Calculate instantaneous value of modulus of the Pointing’s vector (vector of energy fluxes density of EMW):

P (x, t)= E_y (x, t)× H_z (x, t)= E_m× cos(w t – kx +j₀)× H_m× cos(w t – kx +j₀)= E_m× H_m× cos²(w t – kx +j₀).

Let’s check whether the obtained formula gives the unit of energy fluxes density [ W / m² ]

;

Let’s substitute the numerical values:

P (x, t)= 2, 5 × 0, 02 × cos²(2× 10⁶p× t –0, 02p× x –p / 2).

At the moment of time of t ₁=1, 5 T =10 ^–6 s (given by the problem statement) and at the point with coordinate x ₁ = 1, 25l= 1, 25× 2p / k; then x ₁ =1, 25× 2p / 0, 02p=125 m we obtain:

P (x ₁, t ₁) = 2, 5 × 0, 02 × cos²(2× 10⁶p× 1, 5× 10 ^–6 – 0, 02p× 125– p / 2) =

= 0, 05 × cos²(3p – 2, 5p– p / 2) = 0, 05 × cos²(0, 5p– p / 2)= 0, 05 × (1)²=50 mW / m².

Let’s plot it on the graph obtained Pointing’s vector (see Fig. 3.6, point x = x ₁ = 1, 25l).

4) The intensity of electromagnetic wave is the average energy in time, going through the unit plane, which is perpendicular to the direction of wave propagation;

,

where Р – average value of vector modulus of energy fluxes density of EMW (modulus of Pointing’s vector).

Let’s make the calculations: I = 0, 5 · 2, 51 · 0, 02= 2, 51 · 10^-2 W / m² = 25 mW / m².