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S. Persheyev, D.A. Rogatkin
80
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5
Fig. 1.
Radiant flux
F
(
x
)
inside the
continuous material sample and a
formation of the flux decrement
The Bouguer's law initially was proved and confirmed in experiments, but we are
interested in a theoretical derivation of the exponential law. For this purpose, one
should
assume
the difference Δ
F
between
F
(
x +
Δ
x
) and
F
(
x
)
.
For example:
(
) ( )
( ) ,
F F x x F x KF x x
(1)
where
K
is the absorption coefficient. Its dimension is [
x
]
−1
, i. e., the inverse dimen-
sion to a distance dimension. In the general case,
K
is a function of wavelength
, i. e.
K
=
K
(
). Hereinafter, for brevity and clarity, the dependence of the wavelength
are
omitted. Equation (1) claims and defines the linear absorption of
F
(
x
) inside Δ
x
with
the coefficient of absorption
K =
const. This coefficient can be not a constant, but a
function of a coordinate
x
for non-homogenous media. However, it is not a case for
our consideration here. We consider just the simple case of the perfect homogenous,
continuous medium, i. e.
K =
const. According to the classic mathematical definition
of the derivative
0
( ) lim
( ).
x
dF x
F KF x
dx
x
(2)
Therefore, ignoring boundary processes at
x =
0, one can obtain the following
exponential law at
K
= const:
0
( )
.
Kx
F x F e
(3)
In the case of RTE, the more accepted notation for
K
is
a
. As one can see, the
main ''first principle'' in the Bouguer's law is the linear approximation in Eqs. (1) and
(2). It tells us nothing about the inner structure as well as about the inner properties
of the medium, with the exception of a uniformity of them along the
X-
axis. For ex-
ample, if the substance of the sample is a medium with discrete absorbers (Fig. 2),
each of which absorbs the part
a
(
a
< 1) of the radiant flux
F
(
x
), and between absorb-
ers there is no absorption at all, then
(
) ( ) ( )(1 )
( ) ( )((1 ) 1),
N
N
F F x x F x F x a F x F x
a
(4)
were
N
is a number of absorbers ''
a
'' inside Δ
x
. Introducing the absorbers density
,
once again with the dimension [
x
]
−1
, substituting
N x
into Eq. (4), and then
Eq. (4) in Eq. (2), and revealing the infinity in Eq. (2) with the use of L'Hopital's rule,
the limit in Eq. (2) can be evaluated as follows:
0
( ) lim [
ln(1 )] ( ).
x
dF x
F
a F x
dx
x
(5)