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Chapter 2: EM Waves and properties
Waves, a reminder
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Trough or
deep
Wave is a disturbance propagating though a medium. The disturbance moves, but the
medium itself does not. Physical waves come in two
varieties: transverse and longitudinal and are characterized by three
parameters: amplitude, frequency, and wavelength.
Both varieties are described by the same periodic function ๐ ๐ฅ,๐ก = ๐(๐ฅ ยฑ ๐ฃ๐ก), where ๐ฃ is the propagation velocity of the wave (disturbance/perturbation).
https://phet.colorado.edu/en/simulations/category/physics/sound-and-waves
Wave characteristics (1):
https://phet.colorado.edu/en/simulations/category/physics/sound-and-waves
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๐
๐
๐ฅ๐กโฒ
๐ฅ๐ก
๐ฅ
๐โฒ
๐ฃ
๐ฃ๐ก
๐๐ ๐๐
The disturbance ๐ (its nature is not important) which moves in the positive ๐ฅ direction with a constant velocity ๐ฃ, and must be a function of both positions ๐ฅ and time ๐ก and can be expressed as ๐ = f x, t . The shape of ๐ at any instance, say at ๐ก = 0, can be found by holding time constant at that value, i.e.:
๐ = ศf x, t ๐ก=0=f x, 0 = f x
Representing the shape or profile of the wave at that time point.
Wave characteristics (2):
๐ฅ๐กโฒ
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Propagation of a pulse on a spring. The section of
the spring moves up and down as the pulse travels
from left to right.
In a way, the process is analogous to โtakingโ a
โphotographโ of the disturbance as it travels by.
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Wave characteristics (3):
Hence, for instance, ๐ ๐ฅ, ๐ก = ๐0๐โ(๐ฅโ๐ฃ๐ก)2 is a bell-shaped wave, traveling in the
positive ๐ฅ direction with a speed ๐ฃ.
Relating to ๐ (not deformed through space) in coordinate system ๐โฒ, which travels with the pulse at a speed ๐ฃ, ๐ is no longer a function of time, and as we move along with ๐โฒ, we see a stationary constant profile with the same functional form as at๐ก = 0, i.e. ๐ = ศ๐ x, t ๐ก=0=๐ x, 0 =๐ x but for xโฒ, ๐ = ๐ xโฒ
According to the figure above, xโฒ= x๐ก -๐ฃ๐ก. Hence ๐ can be written in terms of the variables associated with the stationary ๐ system as:
๐ ๐ฅ, ๐ก = ๐(๐ฅ โ ๐ฃ๐ก)
This then, represents the most general form of one-dimensional wave function.
It should be emphasized that one should only choose the shape (function), say
j(x), and then substitute (๐ฅ โ ๐ฃ๐ก) for ๐ฅ in j(x), to make it a wave.
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Wave characteristics (4):
Does ๐ ๐ฅ, ๐ก = ๐ ๐ฅ ยฑ ๐ฃ๐ก = f u indeed addresses the partial wave function?
To prove that, we apply the chain rule for derivation [i.e. ๐๐ฆ
๐๐ฅ=
๐๐ฆ
๐๐ข
๐๐ข
๐๐ฅ]. In our case:
๐๐ข
๐๐ฅ= 1 and
๐๐ข
๐๐ก= ยฑ๐ฃ.
Then:๐๐
๐๐ฅ=
๐๐
๐๐ข
๐๐ข
๐๐ฅ=
๐๐
๐๐ขand
๐๐
๐๐ก=
๐๐
๐๐ข
๐๐ข
๐๐ก= ยฑ๐ฃ
๐๐
๐๐ข
Next, taking the second derivatives, one gets:
Combining both results, a and b, to eliminate ๐2๐
๐๐ข2, we get:
๐2๐
๐๐ฅ2=
1
๐ฃ 2๐2๐
๐๐ก2
Proving that ๐ ๐ฅ, ๐ก = ๐ ๐ฅ ยฑ ๐ฃ๐ก is a solution of the partial differential wave equation, independent of the form of the function ๐.
๐2๐
๐๐ฅ2=
๐
๐๐ข
๐๐
๐๐ฅ
๐๐ข
๐๐ฅ=
๐2๐
๐๐ข2[a] ๐๐๐
๐2๐
๐๐ก2=
๐
๐๐ข
๐๐
๐๐ก
๐๐ข
๐๐ก= ยฑ๐ฃ โ
๐2๐
๐๐ข2โ ยฑ๐ฃ = ๐ฃ 2
๐2๐
๐๐ข2[b]
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Wave characteristics (5): The vector significance of the wave number เดค๐
This yields a plane to which เดค๐ must be perpendicular to and าง๐๐ is the position of each point ๐๐ on this plane in respect to a given origin).
๐
๐๐
๐๐
Plane wavefront, เดฅ๐, เดฅ๐๐ and ๐ โ เดค๐ = constant
A wave front (upper figure) is a surface on
which points ๐๐ are affected in the sameway by a wave at a given time.
In other words: the surface on which points have
identical phases, i.e. ฯ = เดค๐ โ าง๐ ยฑ ๐๐ก = ๐๐๐๐ ๐ก๐๐๐ก [like
potential surface in electrostatics] for a given time
point.
Wavefront for a harmonic plane wave
A representative wave ๐ ๐ and corresponding plane wave fronts are given in the lower figure.
The shortest form of the equation of a plane perpendicular
to เดฅ๐ is: เดฅ๐ โ เดค๐ = constant, i.e.
The product เดค๐ โ าง๐๐ = ๐๐๐๐๐ ๐๐ must be the same for every าง๐๐ (เดค๐ ๐๐ ๐๐๐๐ ๐ก๐๐๐ก).
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Wave fronts: plane (a), cylindrical (b) and spherical (c ). A-F 699
๐
Wave characteristics (6): Various shapes of waveforms
1. Plane electromagnetic wave in an unbound medium
1.1 Plane Wave in a Simple, Source-Free ๐, ๐ฝ, ๐, โณ = 0 and Lossless Medium
Where ๐ is the volume density of free net charge, ๐ฝ is the current surface density, ๐ is
the polarization vector in dielectric (Coulombsโ ๐โ2) and โณ is the volume Magnetization density vector (Ampereโ ๐โ1) in magnetic medium.
1.1.1 ๐ป ร ๐ธ= -๐๐ต
๐๐ก
1.1.3 ๐ป ร ๐ต= - ๐0๐0๐๐ธ
๐๐ก
1.1.2 ๐ป โ ๐ธ= -๐
๐0
1.1.4 ๐ป โ ๐ต = 0
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Our starting point consists of the 4 differential Maxwell equations:
The two curl Maxwellโs equations indicate the fact that changing Magnetic Field with
time (1.1.1) produces Electric Fields and vice versa (1.1.3) and hence, necessarily
lead to propagation of electromagnetic waves.
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Taking the curl of 1.1.1 and substituting it into the right side of 1.1.3 gives:
๐ป ร ๐ป ร ๐ธ = โ๐
๐๐ก๐ป ร ๐ต = โ
๐
๐๐ก( โ ๐0๐0
๐๐ธ
๐๐ก) =๐0๐0
๐2๐ธ
๐๐ก2
Vector analysis teaches that ๐ป ร ๐ป ร ๐ธ = ๐ป(๐ป โ ๐ธ) + ๐ป2๐ธ = ๐ป2๐ธ since ๐ป โ ๐ธ = ๐ = 0 one gets:
๐ป2๐ธ = ๐0๐0๐2๐ธ
๐๐ก2[2]
We identify Equations 2 and 3 as differential wave equations for ๐ธ and for ๐ต
Similarly, taking the curl of 1.1.3 and substituting Eq. 1.1.1 into itโs right side gives:
๐ป ร ๐ป ร ๐ต = โ๐0๐0๐
๐๐ก(๐ป ร ๐ธ)= โ ๐ธ๐. 1.1.1 = โ๐0๐0
๐
๐๐ก(โ๐๐ต
๐๐ก) = ๐0๐0
๐2๐ต
๐๐ก2
Again, ๐ป ร ๐ป ร ๐ต = ๐ป(๐ป โ ๐ต) + ๐ป2๐ต = ๐ป2๐ต and, since ๐ป โ ๐ต = 0 , one gets:
๐ป2๐ต = ๐0๐0๐2๐ต
๐๐ก2[3]
1.1.1 ๐ป ร ๐ธ= -๐๐ต
๐๐ก
1.1.3 ๐ป ร ๐ต= - ๐0๐0๐๐ธ
๐๐ก
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Hence in Vacuum, Maxwell Equations teaches that each of the Cartesian (scalar) components of ๐ธ and ๐ต obeys the three-dimensional wave equation:
6 2 12 1 1 1 2 2 20 0
9 8
1 1
1.2566 10 8.8541878 10 ( sec )
0.29979 10 3 10 300,000sec sec sec
cm kg C N kg m m C
m m km
Maxwellโs speculation : โThis velocity is so nearly that of light (measured by Fizeau (1849): 313,300
km/second, M.D.), that it seems we have strong reason to conclude that light itself (including radiant heat,
and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through
the electromagnetic field according to electromagnetic laws.โ
Further more, Maxwell Equations imply that:
1. electromagnetic waves indeed propagate in vacuum and
2. these waves travels at the speed c:
22
2 2
1 ff
v t
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History intermezzo:
In 1849, Fizeau calculated a value for the speed of light to a better precision than
the previous value determined by Ole Romer in 1676. He used a beam of light
reflected from a mirror 8 kilometers away. The beam passed through the gaps
between teeth of a rapidly rotating wheel. The speed of the wheel was increased
until the returning light passed through the next gap and could be seen.
Hippolyte Fizeau 1819-1896
James Clerk Maxwell 1831โ1879
Maxwell: โThis velocity is so nearly that of light, that it seems we have strong reasonto conclude that light itself (including radiant heat, and other radiations if any) is anelectromagnetic disturbance in the form of waves propagated through theelectromagnetic field according to electromagnetic laws.โ
Einstein on Maxwellโs work: โmost profound and the most fruitful that physics has experienced since the time of Newton .โ
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Many of the problems of mathematical physics involve the solution of partial differential equations. In electromagnetics, these can be generally divided into two types of second order partial differential equations:
Poissonโs equation: ๐ต๐๐ = ๐(๐,๐,๐), where u may present the same physical quantities listed for Laplaceโs equation, but in regions containing matter/electric charges, etc. The function ๐ ๐,๐,๐ is called โthe source densityโ, for instance in electricity it is related to ๐๐ .
Laplaceโs equation: ๐ต๐๐ = ๐, where the function u might describe the gravitational/electrical potential functions in no-matter/charge region and steady state temperature in a non-heat source region as well.
Diffusion of heat flow equation: ๐ต๐๐ =๐
๐ถ๐๐๐
๐๐, where u may present non-steady state temperature in a
non-heat source region or the concentration of diffusing material. ๐ผ is a constant which is defined as the diffusivity.
Wave equation: ๐ต๐๐ =๐
๐๐๐๐๐
๐๐๐, where u may present the displacement from equilibrium of (a) vibrating
string/membrane, or (in acoustics) the vibrating medium (gas, liquid, solid), of (b) the electrical current or
potential along a transmission line and of (c) the components ๐ธ and ๐ต of an electromagnetic wave.
Partial differential equations, a reminder:
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We will analyze the case of the differential wave equation for ๐ธ (Equation 2). First, let us write the full expression: (Equation 2):
2 2 2 2 2 22
2 2 2 2 2 2
2 2 2
2 2 2
2
2 2
ห ห( , , , ) ( , , , )
ห( , , , )
1ห ห ห( , , , ) ( , , , ) ( , , , )
x y
z
x y z
E E x y z t x E x y z t yx y z x y z
E x y z t zx y z
E x y z t x E x y z t y E x y z t zc t
So the wave equation independently holds true for each of the components of the vector field เดค๐ธ . For convenience, we shall solve the scalar wave equation for ๐ธ๐:
2 2 2 2
2 2 2 2 2
1( , , , ) ( , , , )x xE x y z t E x y z t
x y z c t
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2 2 2 2
2 2 2 2 2
1( , , , ) ( , , , ) [4]x xE x y z t E x y z t
x y z c t
One way, very much popular in physics, to solve said partial differential equations, is by the method of SEPARATION OF VRIABLES.
๐โฒโฒ ๐ฅ ๐ ๐ฆ ๐ ๐ง ๐(๐ก) + ๐ ๐ฅ ๐โฒโฒ ๐ฆ ๐ ๐ง ๐(๐ก) + ๐ ๐ฅ ๐ ๐ฆ ๐โฒโฒ ๐ง ๐(๐ก) = 1
๐2๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง แท๐ ๐ก [6]
Dividing Eq. 6 by ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง T t ๐ฆ๐๐๐๐๐ :๐โฒโฒ
๐+๐โฒโฒ
๐+ ๐โฒโฒ
๐=
1
๐2
แท๐
๐[7]
Since Eq. [7] holds true for every point in space (each value of x, y and z) and for every time point, each of the components of Eq. 7 must equal constant. This is to say:
๐โฒโฒ
๐= โ๐2;
๐โฒโฒ
๐= โ๐2 ;
๐โฒโฒ
๐= โ๐2 and
1
๐2
แท๐
๐= โ๐2 8
Introducing Eq. 5 into 4 yields:
The basic strategy is: looking for a solution in the form of products of functions, of which each depends on only one of the coordinates. That is to say:
๐ฌ ๐,๐,๐, = ๐ฟ ๐ ๐ ๐ ๐ ๐ ๐(๐ญ) [5]
Rewriting Eq. 7 one gets:
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๐โฒโฒ
๐= โ๐2โน๐โฒโฒ+๐2๐ = 0 ; and similarly ๐โฒโฒ+๐2๐ = ๐ ; and ๐โฒโฒ+๐2๐ = ๐ ; ๐๐๐
1
๐2
แท๐
๐= โ๐2โน ๐โฒโฒ+๐2๐2๐ = 0
Therefore, the solutions of the ratio functions in [8] are to describe harmonic function, i.e. sines, cosines, and their combination. We choose the following
X=๐๐๐๐ฅ ; Y=๐๐๐๐ฆ ; Z=๐๐๐๐ง and T=๐๐๐๐๐ก. Consequently,
๐ฌ ๐, ๐, ๐, = ๐ฟ ๐ ๐ ๐ ๐ ๐ ๐(๐ญ)= ๐ธ0๐๐(๐๐ฅ+๐๐ฆ+๐๐งโ๐๐๐ก) = = ๐ธ0๐
๐ (๐๐ฅ เท๐ฅ+๐๐ฆ เท๐ฆ+๐๐ง ฦธ๐ง โ ๐ฅ เท๐ฅ+๐ฆ เท๐ฆ+๐ง ฦธ๐ง โ๐๐๐ก] =
= ๐ฌ๐๐๐(เดฅ๐โเดค๐โ๐๐๐), where ๐ โ เดค๐ โก ๐๐ฅ เท๐ฅ + ๐๐ฆ เท๐ฆ + ๐๐ง ฦธ๐ง
Since the multiplicity ๐๐๐ must result in an angle (in radians), i. e. ๐๐๐=๐๐ก โน ๐ =๐
๐=
2๐๐
๐๐โน ๐ =
2๐
๐and ๐ =
2๐
๐
where T is defined as the cycle (period) time.
We choose the constants to be negative: โ๐2 and not positive +๐2 , since the latter results in a nonphysical solution:
๐โฒโฒ
๐= เต
+๐2โน ๐2 = ๐2; ๐ = ๐ โน ๐ ๐ฅ โ ๐๐๐ฅ, ๐ ๐๐๐ ๐โ๐ฆ๐ ๐๐๐๐ ๐ ๐๐๐ข๐ก๐๐๐
โ๐2โน ๐2 = โ๐2; ๐ = ยฑ๐๐ โน ๐ ๐ฅ โ ๐ยฑ๐๐๐ฅ, ๐ ๐โ๐ฆ๐ ๐๐๐๐ ๐ ๐๐๐ข๐ก๐๐๐
or ๐ฌ ๐, ๐, ๐, can be ๐ฌ๐๐๐๐ เดฅ๐ โ เดค๐ โ ๐๐๐ ; ๐ฌ๐๐๐๐ เดฅ๐ โ เดค๐ โ ๐๐๐ or a combination of them, where เดฅ๐ โ เดค๐ โ ๐๐๐ โก ๐๐๐๐ ๐๐๐๐๐
โโ < ๐ฅ < โ
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The relation between เดค๐ธ, เดค๐ต, ๐๐๐ เดค๐ ๐๐ ๐ฃ๐๐๐ข๐ข๐ (1):
Taking เดค๐ธ ๐ฅ, ๐ฆ, ๐ง, = เดค๐ธ0๐๐(เดค๐โ าง๐โ๐๐ก), then perfoming the operations ๐ป ร ๐ธ= -
๐๐ต
๐๐กof Eq. 1.1.1 we get:
๐ป ร ๐ธ= ๐ป ร เดค๐ธ0๐๐(เดค๐โ าง๐โ๐๐ก) = ๐๐ข๐๐ ๐ธ๐๐ฅ เท๐ฅ + ๐ธ๐๐ฆ เท๐ฆ + ๐ธ๐๐ง ฦธ๐ง ๐
๐ ๐๐ฅ๐๐ฅ+๐๐ฆ๐๐ฆ+๐๐ง๐๐งโ๐๐ก =
= เท๐ฅ๐๐ธ๐ง
๐๐ฆโ
๐๐ธ๐ฆ
๐๐ง+ เท๐ฆ
๐๐ธ๐ฅ
๐๐งโ
๐๐ธ๐ง
๐๐ฅ+ ฦธ๐ง
๐๐ธ๐ฆ
๐๐ฅโ
๐๐ธ๐ฅ
๐๐ฆ=
= ๐ เท๐ฅ ๐๐ฆ๐ธ๐ง โ ๐๐ง๐ธ๐ฆ + เท๐ฆ ๐๐ฅ๐ธ๐ฅ โ ๐๐ฅ๐ธ๐ง + ฦธ๐ง ๐๐ฅ๐ธ๐ฆ โ ๐๐ฆ๐ธ๐ฅ = ๐เดค๐ ร ๐ธ=-๐๐ต
๐๐ก= ๐๐๐ต
เดค๐ ร ๐ธ = ๐๐ต [9]
From [9] we conclude that in electromagnetic waves in vacuum:
1. ๐ต ๐๐ ๐๐๐๐๐๐๐๐๐๐ข๐๐๐ ๐ก๐ ๐๐๐กโ ๐ธ and เดค๐
2. ๐ธ =๐
๐๐ต =
๐
๐๐ต = ๐๐ต ; ๐. ๐. ๐ฌ ๐๐๐ข๐๐๐๐ ๐ ๐๐๐๐๐ ๐ฉ ๐๐ ๐ฃ๐๐๐ข๐ข๐.
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๐ป ร ๐ต = โ ๐0๐0๐๐ธ
๐๐ก= โ
1
๐2๐๐ธ
๐๐กโถ เดค๐ ร ๐ต =
๐
๐2๐ธ = ๐ ร ๐ต =
๐
๐
๐2๐ธ =
1
๐๐ธ [10]
The relation between เดค๐ธ, เดค๐ต, ๐๐๐ เดค๐ ๐๐ ๐ฃ๐๐๐ข๐ข๐ (2):
Taking เดค๐ต ๐ฅ, ๐ฆ, ๐ง, = เดค๐ต0๐๐(เดค๐โ าง๐โ๐๐ก), then perfoming the operations ๐ป ร ๐ต= - ๐0๐0
๐๐ธ
๐๐กof Eq. 1.1.3 we similarly get:
From [10] we conclude that in electromagnetic waves in vacuum:
1. ๐ธ ๐๐ ๐๐๐๐๐๐๐๐๐๐ข๐๐๐ ๐ก๐ ๐๐๐กโ ๐ต and เดค๐
2. ๐ธ =๐
๐๐ต =
๐
๐๐ต = ๐๐ต ; ๐. ๐. ๐ฌ ๐๐๐ข๐๐๐๐ ๐ ๐๐๐๐๐ ๐ฉ.
3. Conclusion: the vectors ๐ธ ๐ต ๐๐๐ ๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐๐ข๐๐๐ ๐ก๐ ๐๐๐โ ๐๐กโ๐๐, ๐. ๐. ๐๐๐๐๐๐๐ ๐ ๐๐๐โ๐ก โ๐๐๐ ๐ ๐ฆ๐ ๐ก๐๐.
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Except for the amplitudes, are the characteristic constants ๐, ๐, ๐ ๐กโ๐ ๐ ๐๐๐ ๐๐๐ ๐ธ ๐๐๐ ๐๐๐ ๐ต? Is there phase difference between the wave fields?
Referring to ๐๐ต ๐๐๐ ๐๐ต , ๐๐ ๐๐ ๐กโ๐ ๐๐๐๐๐ ๐ค๐๐ฃ๐ ๐ต ๐๐๐ ๐ก๐ ๐๐ธ ๐๐๐ ๐๐ธ ๐๐ ๐๐ ๐กโ๐ ๐๐๐๐๐ ๐ค๐๐ฃ๐ ๐๐ ๐ธ ๐๐๐ ๐๐๐ก ๐๐๐๐ 9 :
เดค๐๐ธ ร ๐ธ = เดค๐๐ธ ร เดค๐ธ0๐๐(เดค๐๐ธโ าง๐โ๐๐ธ๐ก) = ๐๐ต เดค๐ต0๐
๐(เดค๐๐ตโ าง๐โ๐๐ต๐ก+๐ฟ) [11],
๐คโ๐๐๐ ๐ฟ is the phase difference between the waves.
The relation between เดค๐ธ, เดค๐ต, ๐๐๐ เดค๐ ๐๐ ๐ฃ๐๐๐ข๐ข๐ (3):
Next, dividing the left side of Eq. 11 by its right side yields unity (1), and assuming that เดค๐๐ธ ๐๐ ๐๐๐ก ๐๐๐๐๐๐๐๐๐๐ข๐๐๐ ๐ก๐ ๐ธ and there is angle ๐๐ธโ๐ , one gets:
๐ ๐๐๐๐ธโ๐โ๐๐ธโ๐ธ0
๐๐ต๐ต0๐๐ (
เดค๐๐ธโเดค๐๐ โ าง๐โ(๐๐ธโ๐๐ต)๐ก+๐ฟ] โก 1, [12], ๐๐๐ ๐๐ฃ๐๐๐ฆ ๐ก๐๐๐ ๐ก, ๐๐๐ ๐๐ก๐๐๐ าง๐ , ๐ฟ ๐๐๐ ๐๐ธโ๐.
This can occur only when: เดค๐๐ธ = เดค๐๐ = เดค๐; ๐๐ธ = ๐๐ต = ๐, ๐ฟ=0 and ๐ ๐๐๐๐ธโ๐ = 1, ๐. ๐. ๐๐ธโ๐ =๐
2and it turns out that:
๐ ๐๐๐๐ธโ๐โ๐๐ธโ๐ธ0
๐๐ต๐ต0=
๐๐ธ0
๐๐ต0=
1
๐
๐ธ0
๐ต0= 1
0 0 0
2 2 2 2 2 2
0 0 0
2
หห ห ห1 1ห ห4 4 4 4
1.
I dl r dq dll r dq dl r dqdB dt l v r
r dt r c dt dt r c r
v dE E Bc
Ein case v c B
c
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BiotโSavart law
2/27/2020 Electromagnetism 22
Can เดค๐ธ ๐๐ เดค๐ต have a component vibrating in the direction of เดค๐?
Suppose a plane front wave propagating towards เท๐ฅ, ๐. ๐. เดค๐ธ ๐ฅ, ๐ฆ, ๐ง, = เดค๐ธ0๐๐(๐๐ฅ๐๐ฅโ๐๐ก), Therefore, on this plane, where
๐๐ฅ๐๐ฅ=constant, เดค๐ธ ๐๐๐ เดค๐ต ๐๐๐ independent of x and of y, hence the derivatives ๐
๐๐ฆ๐๐๐
๐
๐๐งequal 0. Next, since
div เดค๐ธ = 0 =๐๐ธ๐ฅ
๐๐ฅ+
๐๐ธ๐ฆ
๐๐ฆ+
๐๐ธ๐ง
๐๐ง=
๐๐ธ๐ฅ
๐๐ฅโน ๐ฌ๐ = ๐๐จ๐ง๐ฌ๐ญ๐๐ง๐ญ ๐ข๐ง ๐ฌ๐ฉ๐๐๐. Similarly,
div เดค๐ต = 0 =๐๐ต๐ฅ
๐๐ฅ+
๐๐ต๐ฆ
๐๐ฆ+
๐๐ต๐ง
๐๐ง=
๐๐ต๐ฅ
๐๐ฅโน๐ฉ๐ = ๐๐จ๐ง๐ฌ๐ญ๐๐ง๐ญ ๐ข๐ง ๐ฌ๐ฉ๐๐๐. On the other hand:
= ๐ = ๐
= ๐ = ๐๐ป ร เดฅ๐ธ =
เท๐ฅ เท๐ฆ ฦธ๐ง๐
๐๐ฅ
๐
๐๐ฆ
๐
๐๐ง
๐ธ๐ ๐ธ๐ฆ ๐ธ๐ง
= เท๐ฅ โ 0 โ เท๐ฆ๐๐ธ๐ง
๐๐ฅ+ เท๐
๐๐ธ๐ฆ
๐๐ฅ= โ แถ๐ต๐ เท๐ฅ + แถ๐ต๐ฆ เท๐ฆ + แถ๐ต๐ง ฦธ๐ง โน แถ๐ฉ๐ฟ = ๐ โน ๐ฉ๐ฟ ๐ = ๐ช๐๐๐๐๐๐๐.
Conclusion: ๐ฌ๐ and ๐ฉ๐ are space and time independent and hence, even not zero, cannot be a wave. That is to say the electromagnetic waves are transverse, whereas the electric and the magnetic fields are perpendicular to each other and
both, vertical to the direction of propagation (๐), while in phase and their real amplitudes are related by:
๐ฉ๐ =๐
๐๐ฌ๐ =
๐
๐๐ฌ๐.
The same treatment with ๐ป ร เดฅ๐ต =1
๐2แถเดค๐ธ yields: แถ๐ฌ๐ฟ = ๐ โน ๐ฌ๐ฟ ๐ = ๐ช๐๐๐๐๐๐๐.
2/27/2020 Electromagnetism 23
Replacing the minus signs in [e] and Integration over space ๐ฝ yields:
๐๐,๐ = เถฑ๐ฝ
าง๐ฝ โ เดค๐ธ๐,๐ ๐๐ = โเถฑ
๐
เดฅ๐ป โ๐ เดค๐ต
๐๐ก+ เดค๐ธ โ
๐เดฅ๐ท
๐๐ก๐๐ โ เถฑ
๐
๐๐๐ฃ เดค๐ธ ร เดฅ๐ป ๐๐ [๐]
Generalization of Joule's law, representing the
rate of power dissipated in the volume ๐ฝ.
Loss (negative sign) rate ofโstoredโ electrical and magnetic(static fields) energy within thevolume ๐ฝ. The terms in bracketsare the magnetic and electricenergy densities, ๐ข๐ ๐๐๐ ๐ข๐ ,respectively .
Utilizing divergence theorem this expression equals:
เถป๐
เดค๐ธ ร เดฅ๐ป ๐๐
Conservation of energy dictates that this term must present the flow rate of energy
inward/outward through the surface ๐ enclosing the volume ๐. Hence, the vectorเดฅ๐ธ ร เดฅ๐ป is a measure of the rate of energy flow per unit area at any point on the surface ๐ .
๐๐ท๐ธ,๐ด
๐๐ฝ=โช = โ าง๐ฑ โ เดฅ๐ฌ๐ธ,๐ด = เดฅ๐ฏ โ
๐เดฅ๐ฉ
๐๐+ เดฅ๐ฌ โ
๐เดฅ๐ซ
๐๐+ ๐ ๐๐ เดฅ๐ฌ ร เดฅ๐ฏ [e]
2/27/2020 Electromagnetism 24
k
๐ ๐๐๐๐๐๐๐: ๐ต ๐๐ ๐ ๐ก๐๐๐๐ ๐ ๐๐๐๐๐๐ ๐กโ๐๐ ๐ธ
The electromagnetic wave
2/27/2020 Electromagnetism 25
Wave characteristics (1): โข ๐= the wavelength or the special period of the wave.
โข ๐= the cycle time or the temporal period of the wave.
โข ๐ =๐
๐, velocity (phase velocity) of wave propagation.
โข ๐ =1
๐๐ ๐๐โ1 =
๐
๐, the temporal frequency, i.e. how many periods occurs during a unit time.
โข ๐ =2๐
๐๐ ๐๐โ1 , the angular temporal frequency, i.e. number of radians per period time โ 2๐๐ = 2๐
๐
๐= 2๐๐ ๐ = ๐๐
โข ๐ =2๐
๐, the wavenumber, i.e. number of radians per unit distance. ๐ is the wavelength. The larger ๐, the smaller k.
โข ๐ โก1
๐๐๐โ1 , the special frequency, i.e. number of waves per unite length (how many wavelengths in a 1 ๐๐?)
โข ๐ =๐
๐= ๐๐ =
๐
๐
โข เดค๐ โ าง๐ ยฑ ๐๐ก ๐๐๐ = ๐, ๐๐๐ ๐๐๐๐ ๐๐๐๐๐
โข ๐ธ0 ; ๐ต0 = wave amplitudes
2/27/2020 Electromagnetism 26
๐ = ๐๐ ๐ = ๐โ1 = ๐๐โ1
๐
๐
๐
Figure 1: Wavelength ๐ and time period ๐ of a wave can be measured between any two special or temporal points with the same phase, such as between crests (on top), or troughs
(on bottom), or corresponding zero crossing as shown.
โข ๐ =1
๐๐ ๐๐โ1 ; temporal frequency, i.e. how many times the wave reaches its maximum in a unite time (right figure).
โข ๐ โก1
๐๐๐โ1 ; special frequency, i.e. how many times the wave reaches its maximum in a unite length (left figure).
โข Longitudinal wave: the wave (medium) vibrates in the direction of its propagation.
โข Transverse wave: the wave (medium) vibrates at right angles to the direction of its propagation.
Wave characteristics (2):
0 0 0
2 2 2 2 2 2
0 0 0
2
หห ห ห1 1ห ห4 4 4 4
1.
I dl r dq dll r dq dl r dqdB dt l v r
r dt r c dt dt r c r
v dE E Bc
Ein case v c B
c
2/27/2020 27
BiotโSavart law
2/27/2020 Electromagnetism 28
Energy and momentum in electromagnetic wave
Refreshing expressions: For convenience, we shall examine what happened in RLC electrical circuit (see figure).
๐ = ๐ผ๐ +๐
๐ถ+ ๐ฟ
๐๐ผ
๐๐ก
= ๐ฝ๐น + ๐ฝ๐ช + ๐ฝ๐ณ
Multiplying the equation components by the current ๐ผ one gets the equation for power (๐) :
๐ผ๐ = ๐ผ2๐ +๐ผ๐
๐ถ+ ๐ฟ
๐ผ๐๐ผ
๐๐ก[1]
๐๐๐ฅ = ๐๐๐๐ ๐ + ๐๐ธ + ๐๐ต
๐ธ๐๐ = ๐ผ๐ โ ๐ผ2 ๐ =
๐ผ๐
๐ถ+ ๐ฟ
๐ผ๐๐ผ
๐๐ก[2]
Moving the expression ๐ผ2๐ to the left side of the equality sign of [1] thepower related to the EM๐, i.e. ๐๐ธ+ ๐๐ต, is isolated, namely:
Lets assume that all components has the same cross section S, length ๐ and the same volume ๐, ๐กโ๐๐:
2/27/2020 Electromagnetism
29
๐ผ๐ โ ๐ผ2๐ =๐ผ๐
๐ถ+ ๐ฟ
๐ผ๐๐ผ
๐๐ก[2]
Introducing : ๐ผ = ๐ฝ โ ๐, ๐ฝ = ๐๐ธ, ๐ = ๐๐๐, ๐ =๐๐
๐ =
๐
๐๐ , B =
๐๐ผ
๐โน ๐ผ๐ต =
๐ต๐
๐, ๐ฟ =
๐๐ต
๐ผ=
๐ตโ๐
๐ผ, ๐ถ =
๐๐
๐๐๐๐ ๐ โ ๐ = ๐ ๐๐๐ก๐:
(๐ฝ๐)(๐ธ๐) โถ าง๐ฝ โ เดค๐ธ๐
๐ฝ2๐2๐
๐๐ =๐ฝ2
๐๐ โถ
าง๐ฝ โ าง๐ฝ
๐๐
๐ /๐๐ธ
๐ผ๐
๐ถ= ๐ผ๐ =
๐(๐๐ ๐)
๐๐ก๐ธ๐ = ๐ แถ๐ธ๐ธ ๐๐ = แถ๐ท๐ธ๐ โถ เดค๐ธ โ แถเดฅ๐ท๐
๐ต๐
๐ผ๐ผ๐
๐๐ก(๐ต๐
๐) โถ เดค๐ต โ แถเดฅ๐ป๐
Rewriting [2] with the new โfieldsโ expressions:
าง๐ฝ โ เดค๐ธ๐๐ฅ๐ก๐ โ๐ฝ2
๐๐ = เดค๐ธ โ แถเดฅ๐ท๐ + เดค๐ต โ แถเดฅ๐ป๐ [3]
Dividing [3] by the volume ๐ , we get the โThevolume power densityโ (VPD) (i.e. the work done
per unit time per unit volume) equation forelectric and magnetic fields and currents:
าง๐ฝ โ เดค๐ธ๐๐ฅ๐ก โ๐ฝ2
๐= เดค๐ธ โ แถเดฅ๐ท + เดค๐ต โ แถเดฅ๐ป โก
๐2๐ค๐ธ๐๐๐ก๐๐ฝ
=โช [4]
Input power โ loss of power = โช , the PD of E and M fields --- all per unit volume
2/27/2020
Electromagnetism 30
Next, Remembering that าง๐ฝ = ๐ เดค๐ธ๐๐๐ก๐๐ , we may write:
าง๐ฝ = ๐( เดค๐ธ๐๐ฅ+เดค๐ธ๐ + เดค๐ธ๐)/โ ฮคาง๐ฝ ๐
โก เดค๐ธ๐, ๐โนโ าง๐ฝ โ เดค๐ธ๐,๐ = าง๐ฝ โ เดค๐ธ๐๐ฅ โ
๐ฝ2
๐[a]
๐๐๐ ๐ ๐๐๐๐ข๐ก โ๐๐๐ก
Recalling าง๐ฝ of Maxwellโs 4th equation:
๐ป ร เดฅ๐ป = าง๐ฝ +๐เดฅ๐ท
๐๐กโน โ าง๐ฝ = โ๐ป ร เดฅ๐ป +
๐เดฅ๐ท
๐๐ก[๐] Introducing าง๐ฝ ๐๐ ๐ ๐๐๐ก๐ โ าง๐ฝ โ เดค๐ธ๐,๐ of a yields:
โ าง๐ฝ โ เดค๐ธ๐,๐ = โเดค๐ธ โ ๐ป ร เดฅ๐ป + เดค๐ธ โ๐เดฅ๐ท
๐๐ก[๐] Next, recalling from vector analysis that:
26 : ๐๐๐ฃ( เดค๐ธ ร เดฅ๐ป) = เดฅ๐ป โ ๐ป ร เดค๐ธ โ เดค๐ธ โ ๐ป ร เดฅ๐ป [๐] and substituting โเดค๐ธ โ ๐ป ร เดฅ๐ป of [d] in [c], one gets:
โ าง๐ฝ โ เดค๐ธ๐,๐ = โเดฅ๐ป โ ๐ป ร เดค๐ธ + เดค๐ธ โ๐เดฅ๐ท
๐๐ก+ ๐๐๐ฃ เดค๐ธ ร เดฅ๐ป ; ๐๐๐ ๐ ๐๐๐๐ ๐ป ร เดค๐ธ = โ
๐ เดค๐ต
๐๐ก๐๐๐ฅ๐ค๐๐๐ โ ๐น๐๐๐๐๐ฆ ๐๐๐ค , the total loss of power per unit
volume d๐ข๐ ๐ก๐ เดฅ๐ธ ๐๐๐ เดค๐ต ๐๐ :
๐๐ท๐ธ,๐ด
๐๐ฝ=โช = โ าง๐ฑ โ เดฅ๐ฌ๐ธ,๐ด = เดฅ๐ฏ โ
๐เดฅ๐ฉ
๐๐+ เดฅ๐ฌ โ
๐เดฅ๐ซ
๐๐+ ๐ ๐๐ เดฅ๐ฌ ร เดฅ๐ฏ [e]
โ๐ฝ2
๐= าง๐ฝ โ ( เดค๐ธ๐๐ฅ+เดค๐ธ๐ + เดค๐ธ๐) = าง๐ฝ โ เดค๐ธ๐๐ฅ + าง๐ฝ โ ( เดค๐ธ๐ + เดค๐ธ๐) = าง๐ฝ โ เดค๐ธ๐๐ฅ + าง๐ฝ โ เดค๐ธ๐,๐ โ
2/27/2020 Electromagnetism 31
๐ ๐๐๐๐ เดฅ๐ป โ๐ เดค๐ต
๐๐ก=
เดค๐ต
๐0
๐ เดค๐ต
๐๐ก=
1
2๐0
๐ เดค๐ต 2
๐๐ก=
1
2
๐0๐ เดฅ๐ป2
๐๐กand เดค๐ธ โ
๐เดฅ๐ท
๐๐ก=
1
2
๐0๐( เดค๐ธ)2
๐๐ก, the first expressions in the left side of
the equilibrium sign of [f] can be rewritten to yield the following equation [g]:
๐๐,๐ = เถฑ๐ฝ
าง๐ฝ โ เดค๐ธ๐,๐ ๐๐ = โ๐
๐๐กเถฑ
๐
1
2๐0 เดฅH
2 +1
2๐0 เดคE
2 ๐๐ โ เถฑ
๐
๐๐๐ฃ เดค๐ธ ร เดฅ๐ป ๐๐ [๐]
though the quantities ๐ข๐ =1
2๐0 เดฅH
2 ๐๐๐ ๐ข๐ =1
2๐0 เดฅE
2 are known to present electric and magnetic energy densities for
static fields (i.e. within a condenser or a coil). However, based on the fact that the integrands in Eq. [g] are defined at agiven point, these quantities fairly represents the stored energy densities in the case of time-varying fields, as well. That is to say, that the correct amount of total electromagnetic energy density, ๐ข๐๐, is always obtained by assigning an amount:
๐ข๐๐ = ๐ข๐ + ๐ข๐ =1
2๐0 เดฅH
2 + ๐0 เดคE2 =
1
2าง๐ต โ เดค๐ป + เดค๐ท โ าง๐ธ [h]
๐๐๐๐๐๐๐๐๐ ๐กโ๐๐ก:๐ป =1
๐๐ต =
1
๐๐๐ธ , than ๐0 เดฅH
2๐๐ ๐ธ๐. โ = ๐0๐ธ2
๐0๐2 =โ [
1
๐0๐2 = ๐0] โ= ๐0๐ธ
2 . Introducing into [h] yield:
๐๐๐ = ๐ข๐ + ๐ข๐ =1
2๐0๐ธ
2 + ๐0 เดฅE2 = ๐บ๐๐ฌ
๐ = ๐๐๐ฏ๐
๐๐,๐ = เถฑ๐ฝ
าง๐ฝ โ เดค๐ธ๐,๐ ๐๐ = โเถฑ
๐
เดฅ๐ป โ๐ เดค๐ต
๐๐ก+ เดค๐ธ โ
๐เดฅ๐ท
๐๐ก๐๐ โ เถฑ
๐
๐๐๐ฃ เดค๐ธ ร เดฅ๐ป ๐๐ [๐]
2/27/2020 Electromagnetism 32
Replacing the minus signs in [e] and Integration over space ๐ฝ yields:
๐๐,๐ = เถฑ๐ฝ
าง๐ฝ โ เดค๐ธ๐,๐ ๐๐ = โเถฑ
๐
เดฅ๐ป โ๐ เดค๐ต
๐๐ก+ เดค๐ธ โ
๐เดฅ๐ท
๐๐ก๐๐ โ เถฑ
๐
๐๐๐ฃ เดค๐ธ ร เดฅ๐ป ๐๐ [๐]
Generalization of Joule's law, representing the
rate of power dissipated in the volume ๐ฝ.
Loss (negative sign) rate ofโstoredโ electrical and magnetic(static fields) energy within thevolume ๐ฝ. The terms in bracketsare the magnetic and electricenergy densities, ๐ข๐ ๐๐๐ ๐ข๐ ,respectively .
Utilizing divergence theorem this expression equals:
เถป๐
เดค๐ธ ร เดฅ๐ป ๐๐
Conservation of energy dictates that this term must present the flow rate of energy
inward/outward through the surface ๐ enclosing the volume ๐. Hence, the vectorเดฅ๐ธ ร เดฅ๐ป is a measure of the rate of energy flow per unit area at any point on the surface ๐ .
๐๐ท๐ธ,๐ด
๐๐ฝ=โช = โ าง๐ฑ โ เดฅ๐ฌ๐ธ,๐ด = เดฅ๐ฏ โ
๐เดฅ๐ฉ
๐๐+ เดฅ๐ฌ โ
๐เดฅ๐ซ
๐๐+ ๐ ๐๐ เดฅ๐ฌ ร เดฅ๐ฏ [e]
2/27/2020 Electromagnetism 33
Motti: Explain the idea of vector Poynting and relate to the case of static electromagnetics.
๐โ๐ ๐ฃ๐๐๐ก๐๐๐ เดค๐ธ ร เดฅ๐ป ๐๐ ๐กโ๐ Poynting ๐ฃ๐๐๐ก๐๐ เดฅ๐. It is named after its discoverer John Henry Poynting who first derived it in1884. ๐ผ๐ก represents the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic
field, i.e. tโ๐ ๐๐๐ค๐๐ ๐๐๐๐๐๐๐ ๐๐๐๐ ๐๐ก๐ฆ ๐๐ ๐ ๐ก๐๐๐ฃ๐๐๐๐๐ ๐๐๐๐๐ก๐๐๐๐๐๐๐๐ก๐๐ ๐ค๐๐ฃ๐:
๐๐ท๐ธ๐=๐2๐๐ธ๐๐๐ ๐๐ก
=๐๐๐ธ๐๐๐
= เดฅ๐ โก เดค๐ธ ร เดฅ๐ป
John Henry Poynting (1852โ1914)
๐๐๐๐๐๐๐๐๐ ๐กโ๐๐ก:๐ป =1
๐๐ต =
1
๐๐๐ธ, then:
The MKS unit of the Poynting vector is watt per square meter, เดฅ๐ ๐๐พ๐= ๐๐โ2.
เดฅ๐ = เดค๐ธ ร เดฅ๐ป =1
๐๐๐ธ 2 = ๐บ๐๐๐
๐ = ๐๐๐๐ฏ๐
๐ = ๐0๐๐ธ2 = ๐ถ 2๐๐๐คโ1๐โ2๐๐ ๐๐โ1
๐ฃ๐๐๐ก2
๐2= ๐ถ 2๐๐๐คโ1๐โ2๐๐ ๐๐โ1
ฮค๐ฝ๐ข๐๐๐2 ๐ถ 2
๐2=
= ๐๐๐คโ1๐โ2๐๐ ๐๐โ1๐๐๐ค2๐2
๐2=๐๐๐ค ๐ ๐ ๐๐โ1
๐2=๐ฝ๐๐ข๐๐ ๐ ๐๐โ1
๐2=๐๐๐ก๐ก
๐2= ๐๐ท
2/27/2020 Electromagnetism 34
Some magnitudes of electromagnetic waves (a):
โข the electromagnetic energy per unit volume: ๐๐พ๐๐
๐๐ฝ= ๐๐๐ = ๐บ๐๐ฌ
๐ = ๐๐๐ฏ๐
โข N, the surface power density is ๐2๐
๐๐ก๐๐ =
๐3๐
๐๐ก๐๐
๐๐
๐๐= ๐ข๐๐
๐๐
๐๐ก= ๐ข๐๐๐ โน ๐ต = ๐ ๐๐๐ ; ๐๐๐=
๐ต
๐= ๐บ๐๐ฌ
๐ = ๐๐๐ฏ๐
โข Irradiation (intensity) I โก ๐ ๐ก = ๐0๐ธ2๐ก = ๐0๐ธ0
2 ๐๐๐ 2 ๐๐ โ ๐ค๐ก ๐ก =1
2๐0๐ธ0
2 โน ๐ฐ โก ๐ต ๐ =๐
๐๐บ๐๐ฌ๐
๐
โข ๐ , the linear momentum per unit volume of electromagnetic waves:
๐๐ญ๐ฃ๐๐๐ข๐๐๐๐ก
= ๐น โน ๐๐ญ๐ฃ๐๐ = ๐น๐๐ก =๐๐
๐๐ฅ๐๐ก =
๐๐
๐โ ๐๐ ๐ข๐๐๐ก ๐ฃ๐๐ =
๐๐๐๐๐
โน ๐ 2 โ ๐ 1 =๐ข2๐โ๐ข1๐
โน
โน ๐น =๐
๐=๐บ๐๐ฌ
๐
๐=๐๐บ๐๐ฌ
๐
๐๐=
เดฅ๐ฌ ร เดฅ๐ฏ
๐๐โโโโถ
๐
๐=๐๐
๐=
๐
ฮค๐ ๐=๐
๐= โ๐
2/27/2020 Electromagnetism 35
Some magnitudes of electromagnetic waves (b):
โข โ, the radiation pressure per unit volume exerted upon any surface exposed to electromagnetic radiation: the amount of energy loss ๐๐ of electromagnetic wave along ๐๐ฅ of propagation is:
โ๐๐
๐๐ฅ๐๐ฅ = โโฑ๐๐ฅ = โ๐๐ ๐๐ฅ = โ๐๐ [๐],
๐คโ๐๐๐ ๐ ๐๐ ๐๐ "effevctive force" ๐๐๐ก๐๐๐ ๐๐๐๐๐ ๐กโ๐ ๐๐๐๐๐๐๐๐ก๐๐๐ ๐๐๐กโ ๐๐๐ ๐ ๐ ๐๐ ๐กโ๐ ๐๐๐๐ ๐ค๐๐ฃ๐ ๐๐๐๐ ๐ ๐ ๐๐๐ก๐๐๐.Consequently, from [๐], the radiation pressure per unit volume โ equales:
โ = โ๐๐ข
๐๐ฅ๐๐ฅ = โ๐๐ข = ๐ข1-๐ข2, ๐คโ๐๐๐ ๐ข = ๐0๐ธ
2 ๐๐ vacuum ๐0๐ธ2 =
๐ฝ๐ข๐๐๐
๐3.
๐โ๐๐๐๐๐๐๐, โ, the radiation pressure per unit volume, is actually the consequential difference between the energy per
unit volume of orthogonally incident and transmitted waves. The more the illuminated media absorbs the incident wave, i.e.
๐ข2 โผ 0 ๐๐๐ (๐๐ข = ๐ข1), the greater the pressure is. On the other hand, the more transparent the media, the lower the
pressure is.
Practically, the radiation pressure per unit volume is evaluated in a case of fully absorbed beam, i.e. ๐ข2 = 0. Then,
โ = ๐ข1 = ๐ข =๐๐ข
๐= ๐๐ = ๐
เดค๐ธรเดฅ๐ป
๐2โ เดฅโ =
เดฅ๐ต
๐
๐0๐ธ2 = ๐ถ 2๐๐๐คโ1๐โ2
๐ฃ๐๐๐ก2
๐2= ๐ถ 2๐๐๐คโ1๐โ2
ฮค๐ฝ๐ข๐๐๐2 ๐ถ 2
๐2= ๐๐๐คโ1๐โ2
๐๐๐ค2๐2
๐2=๐๐๐ค
๐2=๐ โ ๐๐๐ค
๐3=๐ฝ๐ข๐๐๐
๐3
2/27/2020 Electromagnetism 36
Some magnitudes of electromagnetic waves (c):
In the case of fully reflection (say from a mirror, where ๐ข2 = โ๐ข1), then โ = 2๐ข = 2c๐น
If the angle of incidence ๐๐ โ 0, the cosine components of the incidence, reflected and refracted waves (beams) should be considered when calculating the relevant โ ๐๐ , ๐๐ .
Interestingly enough, the PD of the incoming Sun rays, as measured on Earth, is PD = 1.4 โ 103๐๐โ2 = 1.4๐พ๐๐โ2,
hence: ๐ข =๐๐ท
๐= 4.7 โ 10โ6๐ฝ๐๐ข๐๐ โ ๐โ3
Assuming that Earth is a perfect absorber, and taking into account the directional spread of the Sun rays, then
โ, the radiation pressure per unit volume exerted upon Earth surfaces is:
โ โ 1
3๐ข = 1.6 โ 10โ6๐๐๐ค โ ๐โ2
For comparison, the atmospheric pressure is:
1๐ด๐ก = 10+5๐๐๐ค โ ๐โ2