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8/10/2019 Electric Currents & DC Circuits
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Electric Current
and
Direct Current CircuitsWalker, Chapter 21
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Why bother with Electric Potential V,
Capacitance, and Electric Fields? The electric potential difference between point a) and point
b) in a circuit is the force driving current from a) to b)
Capacitance is universal,
Capacitance limits the speed of switching circuits
Capacitance stores energy
Electric Field energy is the energy of waves, light.
2
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Electric Current
3
Whenever there is a net movement of charge, there existsan electrical current.
A current can flow in a wire: usually electrons.
A current can flow in a liquid solution:
For example Na+, and K+ ions across a nerve cellmembrane.
A current can flow in air or free space:electron or ion beam, lightning.
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Unit of measure
of Electric CurrentIf a charge Q moves through a surface A in a time
t, then there is a current I:
The unit of current is theAmpere (A): 1 A= 1Coulomb/sec.
By convention, the direction of the current is thedirection of flow of thepositive charges.
If electrons flow to the left, that is a positive current tothe right.
4
t
QI
e-I
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Resistivity
5
In most materials, in order for a steady current I to flow theremust be an electric field Einside the material. For each chargeq, the electric field Eproduces a force F = qE, this causes the
charge to accelerate, however, due to collisions with thesurrounding medium, there is a viscous force roughlyproportional to the mean velocity of the charges: F = q E-v g. Onaverage, the free charges travel at a constant velocity
proportional to the electric field: F=0, v = q E/ g. If n is the density of free-charges in the material, andA is thecross sectional area of the material, then the current flowing is
I = n A q v = A(nq2/g)E = A E / r r = g/(nq2)
r = Resistivity = instrinsic property of the medium.
V1 V2
V1 > V2
E
e-
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Voltage, Current, and Resistance A block of material has resistivity r, cross sectional areaA,
and lengthL. If there is an electric fieldEin the material,
currentI = A E/r will flow
from higher potential V1 to lower potential V2.
E = (V1-V2)/L Define ResistanceR =rL / A
V1-V2 = I R = Ohms Law.
6
V1 V2E
e-
Resistivity = r Resistivity 0
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Ohms Law
For many materials, the current I is directly proportional to thevoltage difference V.
We define the resistance, R, of such a material to be:
The unit of resistance is Ohms W): 1 W 1 Volt/Amp
Common resistors used in electrical circuits range from a few W toMW (106W).
IfR is constant: doesnt depend on current, or history of current
flow, and only small variation with temperature, atmosphericpressure, etc,
the material is said to be ohmic, and we write Ohms Law:
7
I
VR
IRV
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Wires & Resistors in Circuits
A piece of wire is a resistor.
However, for good conductors like Cu, Al, Au, Ag,
the resistivity is extremely low. When we analyze a circuit containing wires and
other elements (such as light bulbs), the resistanceof the wires is so low that we can [usually] pretend
the wires are perfect conductors. Current can flow in the wire even though the
potential is everywhere the same inside eachseparate piece of wire.
10
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Temperature Dependence and
Superconductivity
11
The resistivity of most materials depends on the temperature.
For most metals, resistivity increases linearly with temperature over asmall range:
This temperature dependence to resistivity can be exploited to build athermometer: Measure the Voltage required to maintain a fixed current inthe resistor. Changes in V measure changes in the temperature of themedium surrounding the resistor.
Some materials (Pb, Nb, Nb3Sn, YBa2Cu3O7) when verycold (3 to 20 K), have a resistivity which abruptly drops tozero. Such materials are then superconductors. Highest
temperature superconductor is T 100K = -173C
)](1[ 00 TT-rr
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Sample Resistivity values
MaterialResistivity r
(Wm)
Thermal
Coefficient (K-1)
Silver (Ag) 1.59 10-8 4.1 10-3
Tungsten (W) 5.6 10-8 4.5 10-3
Nichrome 100 10-8
Graphite ( C ) 350010-8 -5 10-4
Si 2.5 10-3 -70 10-3
Glass > 1010
12
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Resistors in Circuits
In drawing a circuit, the symbol for a resistor is
This zigzag pattern is a visual reminder that thematerial of the resistor impedes the flow of charge,and it requires a potential difference V between thetwo ends to drive current through the resistor.
Current flows from higher value of potential to lowervalue of potential
13
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Simple Battery Circuit
A battery is like a pump
A pump raises fluid by a height h.
A battery pumps charge up to a higher potential.
14
V
+-0
R
I
I
I = V/R
Current is the same everywhere.
Voltage varies from point to
point around loop.
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An Incandescent Light Bulb is a Resistor(but R depends on Temperature T of filament, and
T depends on current I).
15
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Quiz 1: Which Circuit will light the bulb?
A) B) C)
16
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Power in Electric Circuits
17
Recall that resistance is like an internal friction - energy is
dissipated. The amount of energy dissipated when a chargeQ flows down a voltage drop V in a time t is the power P:
P =U/ t=(QV/t) = IV
SI unit: watt, W= AmpVolt = C V/s = J/sFor a resistor, P=IV can be rewritten with Ohms Law V=IR,
P = I2R = V2/R
Power is not Energy, Power is rate of consumption (or production) of energy
Large power plants produce between 100 MW and 1GW of power. Thispower is then dissipated in the resistors and other dissipative circuits in our
electronic appliances, in the resistance of the windings of electric motors, or is
used to charge batteries for later use.
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Energy and Power
Energy Usage: Power times time = Energy
consumed
1 kilowatt-hour= (1000 W)(3600 s) = (1000 J/s)(3600 s) =3.6106 J
Electricity in VA costs about $0.10 per KWhr
My typical household uses 1KW of power, on average.
There are 8800 hours in a yearIn one year, each household consumes 8800 KWhr, or 3.2 1010 J at a cost of (8800 KWhr)($0.10) = $880.
18
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Direct Current (DC) Circuits
A circuit is a loop comprised of elements such as batteries,
wires, resistors, and capacitors through which current flows.Current can only flow around a loop if the loop is continuous.
Any break in the loop must be described by the capacitance ofthe gap, which allows charge to build up as current flows onto
the capacitors.
For current to continue flowing in a circuit with non-zeroresistance, there must be an energy source. Thissource is often a battery. A battery provides a voltage
difference across its terminals.
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Circuits, Batteries, EMF
Batteries and Electromotive Force (emf)
Any device which increases the potential energy of charges which
flow through it is called a source of emf.
The emf is measured in volts and often written as e.
The emf may originate from a chemical reaction as in a battery orfrom mechanical motion such as in a generator.
20
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Simple Battery Circuit
An incandescent light bulb can beapproximated as an ideal resistor(this is a bad approximation,
because most light bulbs have avery strong temperaturedependence to the resistance).
V=IR
5 Watt bulb with 3 V battery:
P= V2 / R
R = V2/P = (3V) 2/(5 AV)
R = 1.8 V/A = 1.8 W.
21
Di t C t (DC) Ci it MORE
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Direct Current (DC) Circuits - MORE
Includes: batteries, resistors, capacitors
Kirchoffs Rules - conservation of charge
(Laws) follow from: (junction rule, valid at any junction)
- conservation of energy
(loop rule, valid for any loop)
With emf (e): constant current can be maintained
charge pump forces electrons to move
in a direction opposite to the electric field
SI unit for emf Volt (V)
No resistance connecting wires of the loop
0I
0V
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Kirchhoffs Rules
Any charge must move around any closed loop with emf
Any charge must gain as much energy as it loses Loss: IR potential drop across resistor Gain: chemical energy from the battery
(charge go reverse direction from e)
Often what seems to be a complicated circuit can be reduced to asimple one, but not always. For more complicated circuits wemust apply Kirchhoffs Rules:
Junction Rule: The sum of currents entering a junction equalsthe sum
of currents leaving a junction. Loop Rule: The sum of the potential difference across all the
elements around any closed circuit loop must be zero.
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A real battery is not an ideal emf.
A simple circuit with a battery and resistor can be graphically
represented as:
r is known as the internal resistance of the battery. Thevoltage on the terminals of the battery is, therefore, V= e-Ir and the current in the circuit is:
rRI
e
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B tt emf i th DC Circuits
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Battery as emfin the DC Circuits
+ terminal at higher potential then - terminal
V=e-Ir Vterminal voltage
rinternal resistancee - equivalent to open-circuit (I=0) voltageI
- + Potential increasesby e
Potential decreasesbyIrr
I
IrV
IrIRerR
I
e
rIRII 22 e
Total power
of emfPower dissipated as joule heat in:
Load
resistor
Terminal voltage:
Emf:
I
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Combining Circuit Elements
Any two circuit elements can be combined in twodifferent ways:
in series - with one right after the other, or
in parallel - with one right next to the other.
Series Parallel
Combination Combination
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Resistors in Series By the conservation of charge,
The same currentIflows throughall three resistors, and through the
battery. Junction rule at a, b, c, d, separately
31
a b
cd
By Energy conservation (electrostatic potential is a function only ofposition in the circuit) potential drop around the loop from a to d
equals the potential gain from d to a:
(Va-Vb)+ (Vb - Vc) + (Vc - Vd) - e = 0 = - e(Va - Vd).
Ohms law:(Va-Vb)=IR1, (Vb - Vc)=IR2, (Vc - Vd) = IR3
e = (Va-Vd) = IReq = I(R1+R2+R3): Req = R1+R2+R3
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Resistors in Parallel By the conservation of charge,
The currentI splits into three (non-equal)branches such that I=I1+I2+I3.
By Energy conservation potential dropacross each resistor is the same (wiresassumed to have zero resistance).
(Va-Vd) - e 0.
32
a
d
Ohms law:
(Va-Vd)=I1R1, (Va - Vd)=I2R2, (Va - Vd) = I3R3
I1 = e/R1, I2 = e/R2, I3 = e/R3 e = I Req = (I1+I2+I3) Req :
e = (e/R1 + e/R2 + e/R3) Req = e Req (1/R1 + 1/R2 + 1/R3)
1/ Req = (1/R1 + 1/R2 + 1/R3)
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Resistors in both Series and Parallel
Combine the first two in parallel to obtainequivalent resistance R/2.
Combine three in series to obtain equivalentresistance R + (R/2) + R = 2.5 R.
33
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What is the Current in each resistor?
35
I1 I2
I3
U Ki kh ff L I1 I
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Use Kirkhoffs Loop
and Junction Laws to
solve for Currents
Junction Law: I1 + I2 = I3 What is the current in the 9.8 W
resistor?
I1 I2
I3
Loop Rule:Left loop: I1 (3.9 W) + I3 (1.2 W) + I1 (9.8 W)12 V = 0
I1 (13.7 W) + I3 (1.2 W)12 V = 0
Right loop: I2 (6.7 W) + I3 (1.2 W) + 09.0 V = 0
Three unknowns, three equations: system is solvable.
Use Junction eqn to eliminate I3 in all other equationsI1 (13.7 W) + (I1 + I2) (1.2 W)12 V = 0 I1 (14.9 W) + I2(1.2 W)12 V = 0
I2 (6.7 W) + (I1 + I2) (1.2 W)9.0 V = 0 I1 (1.2 W) + I2 (7.9 W)9.0 V = 0
Solve left loop for I2 : I2 = [12 V - I1 (14.9 W) ] / (1.2 W)
Substitute into right loop:
I1 (1.2 W) + [12 V - I1 (14.9 W) ] (7.9 W) /(1.2 W)9.0 V = 0
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Final step in substitution--solving
I1 (1.2 W) + [12 V - I1 (14.9 W) ] (7.9 W) /(1.2 W)9.0 V = 0
37
( )( ) ( )
AVI
VI
VVI
VVI
722.09.96/70
709.96
790.91.982.1
2.1
9.7120.9
2.1
9.79.142.1
1
1
1
1
W
-W-
-W-W
W
W-
W
WW-W
Now substitute backwards to get I2 and I3:I2 = [12 V - I1 (14.9 W) ] / (1.2 W)
I2 = [12 V(0.722A) (14.9 W) ] / (1.2 W) = -0.764 AI1 + I2 = I3I3 = (0.722A) -0.764 A
I3 = -0.042 A
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Equivalent Resistance
The current I is the same in both The current may be differentin resistors, so the voltage Vba must each resistor, butthe voltage satisfy: Vba isthe same across each Vba=Va-Vb=IR1+IR2=I(R1+R2)
resistor and the total currentis conserved: I=I1+I2
Req=R1+R2 1/Req =1/R1+1/R2
c
Ci it t i i
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Capacitors are used in electronic circuits. The symbolfor a capacitor is
We can also combine separate capacitors into oneeffective or equivalent capacitor. 2 capacitors canbe combined either in parallel or in series.
Series ParallelCombination Combination
C1 C2 C2
C2
Circuits containing
Capacitors
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Parallel vs. Series Combination
Parallel Series
charge Q1 , Q2 charge on each is Q
total Q=Q1 + Q2 total charge is Q
voltage on each is V voltage V1 + V2= V
Q1=C1V Q=C1V1
Q2=C2V Q=C2V2
Q=CeffV Q= CeffV = Ceff(V1+V2)
Ceff=C1+C2 1/Ceff=1/C1+1/C2
C1 C2
Q -Q Q -Q
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RC Circuits
We can construct circuits with more than just resistor, forexample, a resistor, a capacitor, and a switch:
When the switch is closed the current will not remainconstant.
Capacitor acts as an open circuit: I=0 in branch with capacitor
under study state condition.
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Capacitor Charging
Lets assume that at time t=0, the capacitor is uncharged, andwe close the switch. We can show that the charge on thecapacitor at some later time t is:
q=qmax(1-e-t/RC)
RC is known as the time constant , and qmax is the maximumamount of charge that the capacitor will acquire:
qmax=Ce
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Capacitor Discharging
Consider this circuit with the capacitor fully charged at timet=0:
It can be shown that the chargeon the capacitor is given by:
q=qmaxe-t/RC
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Ammeters and Voltmeters
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Ammeters and Voltmeters
Just as a real battery is not a perfect EMF,
Real ammeters and voltmeters are not perfect either.
An Ammeter can be represented as a small resistance, with a
perfect voltmeter in parallel
A voltmeter can be represented as a large resistance, with a
perfect voltmeter in parallel.What does large or small mean?
Than must be defined in relation to your circuit.
All Ammeters & Voltmeters will distort the circuit
they are trying to measure.
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