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Direct Current Circuits 1
Direct Current Circuits
Direct Current Circuits 2
Electric Current (Charges in Motion)
Electric Current (I)The net amount of charge that passes through aconductor per unit time at any point. Currentis defined as:
!
I =dQdt
Electric current is measured in coulomb’s per second oramperes. (1A = 1 C/s)In a single circuit, the current at any instant is the sameat one point as any other point. (Charge is conserved.)
Direct Current Circuits 3
The free electrons in conductors are in constantrandom motion, so there is no net flow of chargein any one particular direction.
If a steady electric field is established inside aconductor then charged particles are subjected toa steady force .The charges moving in a conductor have frequentcollisions with the massive nearly stationary ionsof the material.
Drift Velocity
!
v E
!
v F = q
v E
Direct Current Circuits 4
The net effect of the electric field is that inaddition to the random motion of the chargedparticles within a conductor, there is also a verysmall net motion or drift of the moving chargedparticles as a group in the direction of the electricforce.
This motion is described in terms of the driftvelocity of the particles.
Drift Velocity
!
v v d
Direct Current Circuits 5
Drift Velocity
!
"Q = NevdA"t
!
I ="Q"t
!
= NevdA
!
N = concentration of charges m"3( )
Direct Current Circuits 6
Current Density
The current per unit cross-section area is called thecurrent density J.
!
J =IA
!
=NevdAA
!
J = NevdThe electric field E in terms of current density J is:
!
v E = "
v J
The constant ! is called the resistivity of the material.
Direct Current Circuits 7
Batteries (emf)
In order to produce an electric current in a circuit, apotential difference is needed. Batteries are one way ofproviding a difference in potential (called electromotiveforce or emf).
_+
"
Direct Current Circuits 8
Schematic Diagrams
The direction of current is by convention thedirection a positive charge moves through thecircuit, which is towards the negative terminalof the battery.
+#
R
I I
II
II
"
Direct Current Circuits 9
Ohm’s LawGeorg Ohm (1787-1854)• Found that the current in a metal wire is proportional
to the potential difference V applied to its ends.• Current also depends upon the conductivity of the
wire.• It is more common to talk about resistance (inverse of
conductivity) and express this relationship as:
!
I =VR
• The unit for resistance is called the ohm and isabbreviated $ (omega)
!
or V = I " R
Direct Current Circuits 10
Ohm’s Law
+#
For power sources:
ab
I
!
"V =Va #Vb
!
=Vab > 0
For resistive loads:
+ #ab
I
!
"VR =Va #Vb
!
=Vab < 0
R"
Direct Current Circuits 11
Terminal VoltageThe potential difference in a real battery is not equal to the emfdue to internal resistance within the battery. This lowers thevoltage available to the circuit.
Vab is called the terminal voltage of the battery.
_ +
I
r
b a
!
Vab ="# Ir"
Direct Current Circuits 12
Resistance (R) and Resistivity (!)It can be experimentally determined that the resistanceof a wire is directly proportional to its length l andinversely proportional to its cross-sectional area A.
!
R = "l
A
The proportionality constant ! is called the resistivityand depends upon the material used for the wire.
!
" [=] # $m
Direct Current Circuits 13
Electric PowerTo find the power transformed by an electric devicerecall that energy is Q%V. Power is the rate energy istransformed in the device or:
!
P =Q"Vt
The SI unit for power is a J/s or watt (1 W = 1 J/s).For resistors, combining the above with Ohm’s Lawresults in:
!
P = I2R
!
="V 2
R
!
= I"V
Direct Current Circuits 14
Measuring Voltage
• Voltmeters are placed in parallel with the pointsbetween which the voltage measurement is made
• Voltmeters have a very high resistance and do notaffect the circuit (they draw a very smallcurrent)
_+R1
R2
R3
!
Vmeter =VR3V
"
Direct Current Circuits 15
Measuring Voltage
• Voltmeters placed in series will block the currentand create an open circuit.
_+R1
R2
R3
!
Vmeter = E
V
"I = 0
_+R1
R2
R3
Direct Current Circuits 16
Measuring Current
• Ammeters are placed in series with the devicethrough which the current measurement is made
• Ammeters have a very low resistance and do notaffect the circuit (the voltage drop is very low)
_+
R2R1
!
Imeter = IR2A
"
Direct Current Circuits 17
Measuring Current
• Ammeters placed in parallel will create a short circuit.Anything parallel to the meter will have no voltage acrossthem and therefore no current
_+
R2R1
"
AI1 = I2 = 0
R3
!
I =ER3
Direct Current Circuits 18
Resistors in Series
R1 R2 R3
I
ab
!
Vab =V1 +V2 +V3
!
Vab = I R1 + R2 + R3( )
!
I =Vab
R1 + R2 + R3
!
=VabRs
!
= IR1 + IR2 + IR3
Direct Current Circuits 19
Resistors in Series
• Current is the same through each resistor and isthe same as the current in the equivalentresistance
• Voltage drop across each resistor is differentunless the resistance is the same
!
Rs = Rii"
R1 R2 R3
I
Direct Current Circuits 20
Resistors in Series (Voltage Divider)
R1 R2
I
ab
!
I =Vab
R1 + R2
!
V1 = IR1
!
V2 = IR2
!
=VabR2R1 + R2
!
=VabR1R1 + R2
Direct Current Circuits 21
Resistors in Parallel
R1
R2
R3
ab
I
!
I = I1 + I2 + I3
I3
I2
I1
!
1Rp
=1R1
+1R2
+1R3
For resistors in parallel the equivalent resistance is
!
=VabRp
!
=VabR1
+VabR2
+VabR3
Direct Current Circuits 22
Resistors in Parallel
• Voltage drop is the same across each resistorand the same as the voltage drop across theequivalent resistance
• Current is different through each resistor, thehigher the resistance the lower the current
!
1Rp
=1Rii
"
R1
R2
R3
Direct Current Circuits 23
Resistors in Parallel (Current Divider)
R1
R2
ab
II2
I1
!
1R
=1R1
+1R2
!
R =R1R2R1 + R2
!
Vab = I " R1R2R1 + R2
#
$ %
&
' (
!
= I " R2R1 + R2
#
$ %
&
' (
!
I2 =VabR2
so
and
and
!
= I " R1R1 + R2
#
$ %
&
' (
!
I1 =VabR1
!
=R2 + R1R1R2
Direct Current Circuits 24
Kirchhoff’s Rules
1.) Junction Rule (Conservation of charge)At any junction point, the sum of all currentsentering the junction must equal the sum of allcurrents leaving the junction.
i1
!
i1 = i2 + i3i2
i3
Direct Current Circuits 25
Kirchhoff’s Rules
2.) Loop Rule (Conservation of energy)The sum of the changes in potential around any closed path of a circuit is zero.
_+R1
R2
R3
!
" +VR1 +VR2 +VR3 = 0
"
Direct Current Circuits 26
Galvanometers• A galvanometer is simply a meter that deflects
in proportion to the current running through it.• The maximum deflection is called the full-scale
deflection.• The key characteristics of a galvanometer are
– The current Ifs required for full-scale deflection.– The resistance Rc of the coil of wire in the meter.
G
!
V = IRc Rc
I
Direct Current Circuits 27
VoltmetersA voltmeter consists of a resistor put in series witha galvanometer. The value of this resistance Rsdetermines the full-scale reading of the meter Vv.
!
Vv = I fs Rs + Rc( )
Rc
Rs
IfsG
!
Rs =Vv " I fs Rc( )
I fsDirect Current Circuits 28
AmmetersAn ammeter consists of a resistor put in parallel(called a shunt resistor or shunt) with a galvanometer.The value of this resistance Rsh determines the full-scale reading of the meter Ia.
!
I fs Rc = Ia " I fs( )Rsh
Rc
Rsh
Ifs
G
!
Rsh =I fs RcIa " I fs( )
Ia
Direct Current Circuits 29
RC Circuits
Direct Current Circuits 30
RC Circuits (Charging)
!
"# iR # qC
= 0
!
i ="R#qRC
_+
R Cb ca
"
Qo = 0
!
"# vab # vbc = 0
_+
R Cb ca
i +q -q
i
"
+ _
Direct Current Circuits 31
!
i =ER"qRC
!
dqdt
=ER"qRC
!
"dtRC
=dq
q"CE
RC Circuits (Charging)
!
dqdt
=CE " qRC
!
dqdt
= "q"CERC
!
dqq"CE
= "dtRC
!
"dtRC0
t# =
dqq"CE0
q#
Direct Current Circuits 32
RC Circuits (Charging)
!
"tRC
= ln q"CE"CE
# $ %
& ' (
!
q(t) =Qf 1" e" tRC( ) where Qf = C #E
!
e"tRC =
q"CE"CE
!
"tRC 0
t= ln q"CE( ) 0
q
!
"CEe" tRC = q"CE
!
q = CE "CEe" tRC
!
"tRC
" 0 = ln q"CE( )" ln "CE( )
!
"dtRC0
t# =
dqq"CE0
q#
Direct Current Circuits 33
RC Circuits (Charging)
!
i =dqdt
The current at any time is given by:!
q(t ) = Qf 1" e" tRC( )
The charge on the capacitor varies according to:
!
= Ioe"tRC
!
="Re#tRC
RC is called the time constant (& ) and is the time ittakes the capacitor to become 63.2% charged.
!
= C" 1# e# tRC( )
Direct Current Circuits 34
RC Circuits (Charging)
!
q(t ) =Q f 1" e" tRC( )
!
i = Ioe" tRC
t
q
Qf
t
i
Io
RC
Io /e
Direct Current Circuits 35
RC Circuits (Charging)
t
q
Qf
Increasing RC
Direct Current Circuits 36
RC Circuits (Discharging)
!
= "dqdt
!
qC" iR = 0
R Cb ca
+Qo -Qoi = 0
!
vC " vR = 0
!
i =qRC
R Cb ca
i +q -q
i
+_
Direct Current Circuits 37
RC Circuits (Discharging)
!
dqqQo
q" = #
dtRC0
t"
!
dqdt
= "qRC
!
ln qQ0
"
# $
%
& ' = (
tRC
!
q =Q0e"tRC
!
ln q( )" ln Q0( ) = "tRC
" 0
!
qQ0
= e"tRC
!
dqq
= "dtRC
Direct Current Circuits 38
RC Circuits (Discharging)
!
= "QoRCe"tRC
The current at any time is given by:
The charge on the capacitor varies according to:
!
q =Qoe" tRC
!
i =dqdt
!
= Ioe"tRC
Direct Current Circuits 39
RC Circuits (Discharging)
!
i = Ioe" tRC
!
q =Qoe" tRC
t
i
Io
RC
Io /e
t
q
Qo
RC
Qo /e
Direct Current Circuits 40
RC Circuits (Discharging)
t
q
Qo
Increasing RC
Direct Current Circuits 41
RC Circuits
!
vC = 0
Uncharged
!
iC " 0shortcircuit
!
vC " 0
!
iC = 0opencircuit
Charged capacitor has same voltage as thedevice that is electrically parallel to it.
!
C
!
C
!
C
!
CFully Charged