§7. Direct electric current and its characteristics.
7.1 Electric current. Direct current.
Current rate and density conditions of direct current (DC)
Terms DC:
1. Avialability at a conductor, which is free electrons.
2. Power supply (electromotive force, difference in potential at the ends of the wire).
If, throug given assumed surface in a conductor total charge is carried wich is not equal to zero, then we say that through this surface electric current flows. It can flow in solids, liquids (electrolytes) in gases.
For current it is necessary that the medium has charged particles that can move within it. These charges are called carriers. Current carriers can be electrons, ions, microscopic particles that are overcharged. The current in the medium occurs if there is electric field inside.
Charge carriers are involved in the molecular thermal motion and they move with velocity V in the absence of the field. When the electric field occurs the chaotic motion of charge carriers is changed into orderly movement with velocity U. Thus, carriers are beginning to move with the velocity:
Since the rate of thermal motion becomes zero, only orderly movement left. Thus, an electric current can be defined as the orderly movement of electric charges. Quantitative characteristic of electric current is the charge that passes through the surface that is considered per unit of to time. This value is called the current rate.
.
(7.1)
Electric current can be stipulated by the positive and negative charges. The transfer of negative charge in one direction is equivalent to transfer the same positive charge in the opposite direction.
.
Electric current can be distributed over the surface on which it runs unevenly. In more details current is characterized by the current density vector, which is numerically equal to current dI, which flows through the point perpendicular to the direction of movement of media surface dS┴
.
(7.2)
The direction of the vector of the velocity of orderly movement is taken as a direction of current density vector.
.
(7.3)
Presume that some unit of volume consist of the concentration of n + n- positive charges and negative charges. Algebraic sum charge equal to the sum of positive and negative e + e- charges. It under the condition of electric field, they will goin some average velocity U + I U-, then for the unit of time through the unit area there pass will be n + U + positive and negative charges nU- that will take n + U + e + and e- nU-positive and negative charges accordingly.
Thus, the expression for the current density can be written as following:
.
(7.4)
Or in the vector form:
(7.5)
.
(7.6)
The latter
is written from equation (7.5), given that
.
Current
that does not change over time, called constant and for
it the following
is true:
.
(7
7)
Then 1 C - is a current that passes through the cross section of the conductor for 1 sec.when current is 1 Amp.
7.2. The equation of continuity
Consider an imaginary closed surface S in a medium through which current flows.
The expression gives the charge, which moves for the time unit from volume V. Limited by surface S.
According to the law of conservation of charge, this value should be equal to the charge reduction rate Q, which is inside this volume.
But
. We will substitute this value of a charge in the previous equation.
.
(7.8)
fig.7.1.
Under the sign of integral there is a partial derivative of density on time, as density can depend not only on time, but also on the coordinate of the considered surface in space. We will transform the left part (7.8) according to Gauss's theorem:
.
(7.9)
The equation (7.9) has to be carried out at any chosen volume on which integration is conducted, and it is possible only if in each point of space the following condition is satisfied:
.
(7.10)
Expression (7.10) is called the continuity equation. It expresses charge conservation law.
In points which are sources of current density vector, there is a reduction of a charge and in that case when current is stationary, potential in various points, density of a charge and other sizes are invariable. Therefore for a direct current of the equation (10):
.
(7.11.)
That means that in case of a direct current when the vector of density of current has no sources, and it means that lines of current don't begin anywhere and don't come to an end anywhere, that is are always closed. That is
=0
d
S
fig 7.2.
7.3. Electromotive force
If in the conductor, where the electric field is created, no action is taken to its support, the movement of current carriers quickly lead to the disappearance of the field in the conductor and the stops flowing.
To maintain current over time it is required from the end of the conductor with less potential (where the media have a positive sign) to continuously carry off the charges that are entrained by the current, to the end with more potential, that do charge cycle, with which they are moving in a closed circuit.
The circulation of the electric field vector is numerically equal to zero. Therefore, in a endless circle, along with areas where positive carriers move downward to decreasing potential, should also be areas where the transfer of positive charges would occur in the direction to increasing potential, i.e. opposite the forces of the electrostatic field.
Moving carriers in these areas is only possible under the virtue of non-electrostatic origin(external forces).
α
1
α 2
fig.7.3.
Thus, to maintain the current outside forces are required that operate around the circumference or on its individual parts. The nature of these forces may be due to chemical processes, diffusion of carriers in a heterogeneous environment or diffusion on the boundary of two different substances, electromagnetic fields (rotor, stator). The foreign forces are characterized by the work they do on charges that move in a circle, and the value is numerically equal to the work of external forces on a single positive charge is called electromotive force (EMF) - ε. This EMF operates in the circle or on the area
.
(7.12)
Given that,
we see that the units of measurement of the EMF in
SI-V (
).
Extraneous force acting on the charge Q can be represented as:
.
(7.13)
This amount of tension * E is called a field intensity of external forces. The work of external forces on the charge Q on the section 1-2:
.
If we divide the work on the by the amount of the charge, we will get the EMF acting in this area.
.
(7.14)
A similar integral calculated in a closed circuit, will EMF, acting in a circle.
.
(7.15)
Thus, EMF of the closed circle can be defined as circulation intensity vector of external forces.
In addition to external forces, the charge is affected by forces of the electrostatic field.
.
Then the resulting force acting at each point on the charge Q:
.
The work done by this force on the charge Q on the area 1-2 is:
.
(7.16)
The value is numerically equal to the work which is carried out by electrostatic and external forces while moving unit positive charge is called a voltage drop or voltage in this area of the circle.
Based on the formula (16) the voltage at the site 1-2
The circuit where outside forces do not operate are called homogeneous, and areas where they operate - heterogeneous. For homogeneous area of the circle tension coincides with the potential difference at its ends.
.
1Ohms is the resistance of a conductor, in which current at the voltage in 1V flows of 1A.
The resistance depends on the shape, size and material of the conductor.
For a homogeneous cylindrical conductor it can be written:
.
where ρ - electrical resistance of the conductor, l - length of conductor, S - cross-sectional area of the conductor.
7.4. Kirchhoff's laws for branched networks
First Kirchhoff's law:
Algebraic sum of the forces of currents that converge at a node is equal to 0. At the,currents entering the unit are considered positive, and the output is negative.
.
The second Kirchhoff's law:
In a closed circuit the algebraic sum of voltages in all areas is equal to the algebraic sum of all the EMF operating in these area.
.
The rules for calculating branching networks:
R1 E
R2
R3
fig. 7.4
1. Choose an arbitrary direction of current forces.
2. According to Kirchhoff's first law write the depence between the record forces of current.
3. We consider the electric circuit by given contours (R1-ε-R2 and R1-ε-R3) and according to the second Kirchhoff’s law write the system of equations.
.
We have a system of three linearly independent equations.
7.5. Capacity of DC
Let us consider an arbitrary area of the DC circuit, to which ends voltage U is applied. For time t through each section of the conductor passes the charge
q = It. This is equivalent to the fact that the charge It is transferred field and the time t from one end to another conductor. This electric field strength and the outside forces that opertes in this area, doing the work
.
(7.18)
Dividing the work on time t, during which it is carried out, we get the capacity which the current develops in this circuit:
.
(7.19)
This capacity can be spent for the works in an area that is considered:on external forces (for this area should move in space), the chemical reactions or by heating of the circuit.
Lecture 8