Electric Currents & DC Circuits

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.

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Electric Current

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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

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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.

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Temperature Dependence and

Superconductivity

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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

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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

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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.

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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).

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Quiz 1: Which Circuit will light the bulb?

A) B) C)

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Power in Electric Circuits

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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.

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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.

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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.

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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

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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.

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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.

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What is the Current in each resistor?

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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

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( )( ) ( )

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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Electric Currents & DC Circuits

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