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






 

When a plane EMW propagates from the source (which is located at point x 0 = 0) along the positive direction of x- axis, the vector of electric field intensity will be changing along the y -axis, and the vector of magnetic field intensity will be changing along the z -axis, according to the equations of EMF:

Ey (x, t)= Em× cos(w tkx +j0); (18)
Hz (x, t)= Hm× cos(w tkx +j0),

where Em and Hm – amplitudes of electric field intensity and magnetic field intensity in a wave correspondingly; j0 – initial phase of the wave source.

Cyclic frequency w [ rad / s ] – is a changing of phase of a wave per second:

w =2p / T =2p f, (19)

here T [ s ] – period is a time of one oscillation of waves’ quantities;

f [ Hz ] – frequency is a number of oscillations of waves’ quantities per second.

Wave number k [ rad / m ] – is a changing of phase a wave per meter:

k = 2p / l, (20)

here l [ m ] – wavelength is a length of one oscillation (distance which is transited by a wave for a period).

Phase velocity of propagation of EMW in medium

, (21)

where speed of light (velocity of propagation of EMW in vacuum):

; (22)

and refractive index

; (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 = Em× Hm / 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 Hz (x, t)= Hm× cos(w tkx –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 tkx –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:

Ey (x, t)= Em× cos(w tkx +j0); Hz (x, t)= Hm× cos(w tkx +j0),

where Em and Hm – 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: Ey (x, t)=2, 5 × cos(2× 106 t –0, 02p× x –p / 2) V /m;

Hz (x, t)=0, 02 × cos(2× 106 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– 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)= Ey (x, tHz (x, t)= Em× cos(w tkx +j0Hm× cos(w tkx +j0)= Em× Hm× cos2(w tkx +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× 106t –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.

Results: Ey (x, t)=2, 5 × cos(2× 106t –0, 02p× x p / 2) V/m;

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

P (x 1, t 1) =50 mW / m2; I = 25 mW / m2.

 

 






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