Volume 02 Issue 11-2022
10
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
11
Pages:
10-19
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I
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(2022:
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ABSTRACT
This is an article on the methodology of teaching physics at the university. The material is devoted to the enumeration
and analysis of physical processes based on mathematics.
KEYWORDS
Physics, mathematics, mathematical model, induction current, Planck's coefficient, frequency, period of light
oscillation.
INTRODUCTION
Research Article
INTER-SUBJECT RELATIONS OF THE COURSE OF GENERAL PHYSICS
WITH THE COURSE OF HIGHER MATHEMATICS
Submission Date:
October 25, 2022,
Accepted Date:
October 30, 2022,
Published Date:
November 07, 2022
Crossref doi:
https://doi.org/10.37547/ajast/Volume02Issue11-03
Saidakhmetov Zamira Vakhabovna
Tashkent Institute Of Chemical Technology, Uzbekistan
Asatov Uralboy Tashniyazovich
Tashkent Institute Of Chemical Technology, Uzbekistan
Bozorov Ismoil Tukhtaevich
Tashkent Institute Of Chemical Technology, Uzbekistan
Fayzullaev Kakhramon Mahmudjanovich
Tashkent Institute Of Chemical Technology, Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 02 Issue 11-2022
11
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
11
Pages:
10-19
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
There is an opinion that since physics is the science of
ideas and experiments, it can be taught without a strict
mathematical apparatus, that a very modest
mathematical background is sufficient to introduce the
basic ideas and explain experiments. We adhere to a
different point of view: in order to master the general
course of physics, the student must not only know the
modern mathematical apparatus, but must also be able
to apply it.
What are the functions of mathematics in the teaching
of physics? Obviously, first of all, the same as in the
science of physics. Mathematics is the language of
physics. As in any activity, a person needs a special
language in order to think and explain something to
others. Mathematics is such a language for physics,
which makes it possible to think in a special,
exceptionally clear and flexible international language.
As R. Feynman pointed out very figuratively,
“Mathematics
is not just another language.
Mathematics is a language plus reasoning, it's like a
language and logic together. Mathematics is a tool for
thinking. It concentrates the results of the precise
thinking of many people. With the help of
mathematics, one statement can be related to
another” (18).
The most common method for studying physical
phenomena by mathematical methods is the modeling
of these phenomena in the form of differential
equations. This is explained by the fact that in order to
compile them, it is sufficient to know only local
connections and information about the entire
phenomenon as a whole is not needed. For example,
when compiling the equations of pendulum
oscillations, we do not start from the seemingly
obvious that the pendulum, taken out of equilibrium,
oscillates, but only use the fact that the restoring force
is proportional to the displacement.
X
k
F
Δ
−
=
The result is an equation whose solution has an
oscillatory character. This solution allows you to
conduct a qualitative and quantitative analysis of the
oscillatory system as a whole. Thus, this mathematical
model makes it possible to study the phenomenon in
general terms, predict its development, and make
quantitative estimates of the changes that occur in it
over time. This is how the undulating propagation of
electromagnetic disturbances was discovered: from
the local properties of the phenomenon to equations,
and from equations to the description of the
phenomenon as a whole.
When studying the section “Physical Foundations of
Mechanics”, it i
s necessary to have an understanding
of the basics of vector algebra, the derivative, the
simplest rules of differentiation, the indefinite and
definite integral, and the ability to integrate the
simplest differential equations. When considering the
rotational motion of a rigid div and gravity,
knowledge of the partial derivative, gradient, scalar,
vector and double vector product of vectors, vector
flux, vector circulation is necessary.
The section "Molecular physics and thermodynamics"
uses the same mathematical apparatus. In addition,
information is needed about the curvilinear integral,
the condition for its independence from the path of
integration, as well as knowledge of the basic concepts
and definitions of probability theory and mathematical
statistics.
When studying electrodynamics, the concepts of
divergence, circulation, and rotor are necessary, and it
is desirable that they be written using the Hamilton
operator, which is widely used in optics, atomic and
molecular physics, and solid state physics. The
Volume 02 Issue 11-2022
12
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
11
Pages:
10-19
SJIF
I
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FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
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Servi
Oscillations and Waves section uses second-order
differential equations as well as functions of a complex
variable.
The final stage of studying the basics of
electrodynamics in the course of general physics is the
study of Maxwell's equations. This means that the
study of electrodynamics in the course of general
physics goes mainly along the inductive path - based on
the analysis of a number of the simplest experimental
facts (electrification of bodies, the interaction of
charged bodies and conductors with current, the
phenomenon of electromagnetic induction, etc.),
certain specific laws are formed, which are then
generalized to Maxwell's equations. Consequently,
here the main path is from the particular to the general,
which, however, does not exclude some elements of
deduction. This path is recommended by the program
and accepted by the vast majority of textbooks for
higher education. Of course, another way is
fundamentally possible - postulating Maxwell's
equations and deriving from them all the provisions of
electromagnetism.
However, this approach seems to be more suitable for
studying physics at a higher level, in particular, in a
theoretical physics course or a special course.
In the inductive approach, Maxwell's equations act as
generalizations of the Ostrogradsky-Gauss theorem for
electric and magnetic field vectors, the law of total
current, and Faraday's law of electromagnetic
induction. In integral form in the International System
of Units, they have the form:
dt
dФ
ds
E
S
S
-
=
(1)
This relation expresses the quantitative relationship between the changing magnetic field
B
and the
vortex electric field
Е
and is one of the basic equations in Maxwell's theory.
When an electric field changes, a magnetic field appears around that field. So the changing electric
field is the displacement current.
So there are three types of currents.
1.
The ordered movement of charges is a current. (In metals, free electrons create a current)
t
q
J
=
2.
Induction current appears when crossing an alternating magnetic field closed circuit.
i
t
Ф
Δ
Δ
-
=
3.
The displacement current is an alternating electric field. An alternating magnetic field creates an
alternating electric field. The bias current density will be:
dt
dD
j
C
=
D
- vector of electrostatic induction. Let's write the law of total current:
полн
S
S
J
dS
H
=
Volume 02 Issue 11-2022
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American Journal Of Applied Science And Technology
(ISSN
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2771-2745)
VOLUME
02
I
SSUE
11
Pages:
10-19
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
- is the length of a closed circuit located inside the conductor, through which alternating current flows. S
is the area of the closed circuit.
полный
J
=
ds
dt
dD
jds
ds
j
S
S
полный
S
∫
∫
∫
+
=
dt
dN
ds
D
dt
d
ds
dt
dD
S
S
=
=
∫
∫
N is the flux of the electrostatic displacement vector.
To mean:
dt
dN
J
J
полный
+
=
dt
dN
J
dS
H
S
+
=
(2)
To these equations, two more equations must be added that express the Ostrogradsky-Gauss
theorem for electric and magnetic fields:
q
dS
D
S
=
(3)
0
=
S
d
B
S
(4)
To this are added the relations between the field vectors:
E
D
0
=
(5)
H
B
0
=
(6)
E
j
=
(7)
Equations (1-7) constitute the system of Maxwell's equations.
They are the most general equations for electric fields.
Note that the quantities
и
,
enter Maxwell's equations as material constants, i.e. as given
quantities characterizing the properties of the medium.
When considering the connection between a conservative force, for example, the force of gravity,
and potential energy, the Hamiltonian differential operator (nabla - operator) can be introduced into the
course of general physics. The elementary work of a conservative force is, by definition, equal to the total
potential energy differential with the opposite sign:
dU
d
F
A
−
=
=
Given that
dz
F
dy
F
dx
F
d
F
z
y
x
+
+
=
and according to the property of the total differential of a function of several variables known to
students
Volume 02 Issue 11-2022
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American Journal Of Applied Science And Technology
(ISSN
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VOLUME
02
I
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11
Pages:
10-19
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
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Servi
dz
z
U
dy
y
U
dx
x
U
dU
+
+
=
we have:
z
U
F
y
U
F
x
U
F
z
y
x
−
=
−
=
−
=
;
;
Or in vector form:
+
+
−
=
+
+
=
z
y
x
z
z
y
y
x
x
e
z
U
e
y
U
e
x
U
e
F
e
F
e
F
F
Note that writing a vector using the notation of orts is
z
y
x
e
e
е
,
,
more convenient than using
orts,
k
j
i
,
,
since
i
is the notation for both the current strength and the imaginary unit, it
j
is the notation
for the current density vector in the course of general physics.
The expressions for the current strength can be symbolically written as:
gradU
U
J
=
−
=
where
z
y
x
e
z
e
y
e
x
+
+
=
Similarly, in electrostatics, the relationship between the strength and potential of the electrostatic
field is derived:
−
=
−
=
grad
E
The correctness of the physical interpretation of a mathematical model can only be established by
direct experience. For example, it became possible to consider Maxwell's equation as a mathematical model
of a real physical process only after Hertz's experimental confirmation of the actual existence of
electromagnetic waves.
The volume and depth of interpretation of this issue in different textbooks for universities differ
significantly. So in the book (3) Maxwell's equation is absent at all; in books (2,5,6) they are given only in
integral form; in book (14) they are expressed in differential form.
Referring to the well-known theorems of Gauss and Stokes, we bring Maxwell's equations to
differential form.
t
B
E
−
=
(8)
t
D
j
H
+
=
(9)
Now it is easy to prove that in a homogeneous and isotropic dielectric in the absence of free charges
and conduction currents, electromagnetic waves are possible, and also to find the speed of these waves.
For this purpose, equations (8 and 9) are reduced to a system with two field vectors:
Volume 02 Issue 11-2022
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VOLUME
02
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11
Pages:
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(2022:
5.
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)
OCLC
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IF
–
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Publisher:
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Servi
t
B
E
−
=
( )
t
E
B
=
0
0
1
Considering that
2
0
0
1
c
=
, we get:
t
B
E
−
=
(10)
t
E
c
B
−
=
2
(11)
Multiply the nabla operator to equation (10) vectorially, i.e., let us differentiate it vectorially with
respect to the coordinates. Since the coordinates and time are independent variables, the operations of
differentiation with respect to them are permutable. Then, using (11) we get:
2
2
2
)
(
)
(
t
E
с
B
t
t
B
E
=
−
=
−
=
(12)
Means:
2
2
2
2
t
E
c
E
=
(I)
Students should point out that exactly the same equation is obtained for the magnetic induction
vector.
2
2
2
2
t
B
c
B
=
(II)
and recommend deriving this equation on your own by applying the vector nabla operator to
equation (11). here it should be shown that the equations obtained are wave equations. This can be done by
assuming that a plane transverse wave propagates in the dielectric.
Graphic images ( I ) and ( II ) will be the following:
Fig. 1
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Fig. 2
Now we will consider what we understand when we say light. There are two theories in the study of
light:
1.
Light consists of small particles (photons) and propagates in portions first in one direction, then
in the other direction. This is Newt's theory.
The energy of one portion of the light beam is:
hv
E
=
Here E is the energy of one portion
h
–
Planck coefficient
v
- frequency
2.
Light is an electromagnetic wave. Maxwell's theory.
Electromagnetic wave equation (mathematical formula of light):
)
(
)
(
2
max
max
+
=
+
=
t
Cos
H
H
t
Sin
E
Е
(one)
Here: E - electric field strength
H - magnetic field strength
The graphical representation of an electromagnetic wave is as follows:
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VOLUME
02
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IF
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E
max
and H
max
are the amplitudes of the electromagnetic wave (light)
- cyclic frequency of electromagnetic wave (light)
t is the time of the electromagnetic wave (light)
2
1
- initial phases of an electromagnetic wave (light)
- wavelength of electromagnetic wave (light).
The speed of light in vacuum is: c=3
.
10
8
m/s=3
.
10
5
km/s.
Light from the sun to the Earth takes 8 seconds.
Light Wavelength:
Т
с
=
c is the speed of light c=3
.
10
5
km/s
T is the period of light oscillation
The propagation of an electromagnetic oscillation in a vacuum or in a medium is an electromagnetic
wave. The electromagnetic wave equation is (1).
The vector diagram of an electromagnetic wave has the following form:
Е
- electric field strength vector.
Н
- the vector of the magnetic field.
c
n
t
T
n
t
T
=
=
=
]
[
]
[
]
[
The period is the time required for one complete oscillation.
Oscillation frequency is denoted by
v
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t
n
=
number of oscillations per unit of time
Frequency unit Hz- Hertz
Гц
c
t
n
=
=
=
1
]
[
]
[
]
[
CONCLUSION
Physics
cannot
be
explained
without
mathematics. Maxwell's equations are the
theorems of Ostrogradsky Gauss, displacement
current, transformation of an electric field into a
magnetic field, a magnetic field into an electric
one, etc. Physical laws are proved by
mathematical equations.
Mathematics can be used to relate one statement
to another.
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1.
V.M. Vergasov Activation of the mental
activity of a student in higher education.
Kyiv, Head publishing house of the
publishing association "Vishcha school"
1971
2.
Detlaf A.A., Yavorsky B.M., Milkovskaya
L.B. Course of physics (For universities) Izd.
4 - e -M. Graduate School 1973
3.
Zisman G.A., Todes O.M. Course of general
physics. T.Z. Optics, physics of atoms and
molecules, physics of the atomic nucleus
and microparticles.
–
M.: Nauka, 1977. -495
p.
4.
S.G. Kalashnikov "Electricity" Publishing
house "Nauka" Moscow 1964 - 668 p.
5.
Kitaygorodsky A.I. Introduction to Physics:
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–
M.: Nauka,
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6.
Matveev A.N. Teaching physics in higher
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–
Kaliningrad,
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7.
Makhmutov M.N. Problem learning. Basic
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8.
Matyushkin A.M. Problem situations in
thinking and learning. M., Pedagogy, 1972
9.
Nasyrov A.Z. Teaching mathematics and
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Vestnik Vyssh. Schools 1975 #3
10.
Pinsky A.A. Methodology as a science.
Owls. pedagogy 1978, no. 12, p. 115-120
11.
Pinsky A.A. Phase and energy relations in
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school, 1980 No. 3, pp73-75
12.
Razumovsky V.G. Development of creative
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1975
13.
Simonov P.V. The brain makes a decision.
–
Science and life, 1974, No. 6
14.
Sivukhin D.V. General course of physics.
T.Z.M, Science -1980
Volume 02 Issue 11-2022
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American Journal Of Applied Science And Technology
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2771-2745)
VOLUME
02
I
SSUE
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Pages:
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SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
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METADATA
IF
–
5.582
Publisher:
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15.
I.V. Savelyev General Physics Course
Volume III Nauka Publishing House
Moscow 1971
16.
S.P. Strelkov Mechanics. State publishing
house Moscow 1956 452 p.
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Skatkin M.N. Improving the learning
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18.
Feynman R. Character of physical laws. Per.
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