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3 – Electric current
Electric circuits consist of electric elements (resistances, condensers, coils, batteries, etc.) connected by wires.
Electric current I through a wire is defined as charge that flows through the cross section of
the wire in a unit of time:
Electric current
t
Q
I
Current in a wire has direction and it can be positive or negative. The direction of the current I
is chosen to be the direction in which positive charges would move. We know that in most of conductors (negatively charged) electrons are moving. Thus the direction of the electric current is opposite to the direction of motion of electrons There is an analogy between the electric current and flow of fluids. Current flowing through a cross section is nothing else than the flux of charge through it.
I
Unit of electric current is Ampere, 1A = 1C/s PHY167 Spring 2021
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Ohm’s law and resistance
Electrons moving through a conductor under the influence of an electric field are colliding with different kinds of obstacles and thus they cannot accelerate unlimitedly. These obstacles resist the electronic motion. As a result, within a short time, electrons acquire a steady velocity. This
means a steady electric current I that depends on the resistance of the material R.
Ohm’s law reads: (^) V RI R V I
or where^ V^ is the potential difference
(voltage) on the resistance R
Unit of resistance is 1 Ohm, denoted as 1 W. Evidently 1 W = 1 V s / C
Resistor Wire Wire
V
Ohm’s law reads: Positive direction of current 2 1 V^ V 2 V 1
If V > 0 then current flows in the
positive direction, I > 0
We will use both V and simply V for the voltage, thus Ohm’s
law can be written as The current flows from high to low potential
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Drude model of the electric resistance
Electric current in a wire can be expressed as (see the next page)
where e is the charge of the electron, n is the concentration of free electrons (number of free
electrons per m^3 ), v is the average velocity of electrons, and S is the cross-section of the
wire. one can check that this formula has a correct unit C/s = A. The electrons are moving under the action of the electric field in the wire and different kinds of obstacles that in the
Drude model are modeled by the drag (liquid friction) force – av, a being the friction constant.
The Newton’s second law for the electron has the form In the stationary state the acceleration is zero, and the driving force from the electric field is balanced by the drag force. From this, one can find the electron’s velocity as
Substituting it into the formula for the current and using E = V/L, one obtains
ଶ Thus we have obtained the Ohm’s law ଶ
5 Cross-section of the wire
- concentration of electrons
Derivation of the formula for the current
The charge crossing the wire’s cross-section during t
(the charge inside the cylinder) is The current:
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Alternating current and its power
Above we have studied the direct current, that is, the current that flows in one direction. Most of electric circuits use the alternating current that typically varies sinusoidally with time (as in the household circuits). The sinusoidal alternating current is described by I(t )I 0 sin( t 0 ) 2 f ,
where I 0 is the amplitude of the current, is the frequency, and 0 is an arbitrary phase. The
power of this current is P I (t )R I sin ( t 0 ) R 2 2 0 2 As the time average of sin^2 (t) is ½, the average power is P I R 2 avr 0 2 1 That can be rewritten in terms of the effective current 0 0 eff 2 eff 0.^707 2 , I I P I R I Similarly one can introduce effective voltage
0 eff 2
eff V
V
R
V
P
where V 0 is the voltage amplitude. Note that the household voltage 110-120 V in North America is the effective voltage.
