chapter 26 october 9, 2012october 9, 2012people.physics.tamu.edu/webb/208_rcw/lectures/lecf... ·...

Direct-Current CircuitsDirect Current Circuits

Chapter 26October 9, 2012October 9, 2012

October 9, 2012 Physics 208 1

Series resistor circuits…

Series Circuit In series circuits, the same current flows through all the components in the i it th t t lcircuit; the total

voltage across the series elements isseries elements is the sum of the voltages across each


o ag s a oss aof the elements.

Parallel resistor circuits..

In a parallel circuit, all components have

Parallel Circuit

the same voltage across them; the t t l ttotal current delivered is the sum of the currentsof the currents through each of the elements.


Now for series network of resistors

Using the definition for R.Rtotal=ΔVtotal/Itotal withtotal total/ total

Vtotal=V1+V2+V3 thenRtotal=(I1R1+I2R2+I3R3)/ItRtotal (I1R1+I2R2+I3R3)/It

thenR l=R1+R2+R3Rtotal=R1+R2+R3.

They just add together…


Parallel networks of resistors..Here, using the definition for Rtotal we find that Rtotal = ΔVtotal/Itotal. With Itotal=V1/R1+V2/R2+V3/R3 then

1/Rtotal=1/R1+1/R2+1/R3…


Combination circuits..


Multiloop circuits and Kirchhoff’s Rules

Kirchhoff’s Rules are based on two conservation laws that we have seenconservation laws that we have seen before in different a context.

Conservation of Charge &gConservation of Energy


Rule # 1…the Junction RuleThe current entering ANY junction in a Current leaving, I1,I2,I3

circuit equals the current leaving that same junctionsame junction

Current entering, I


Rule # 2…the Loop RuleThe sum of the changes in potential around ANY Closed path in a circuit must be zeromust be zero


Our conventions..In solving multiloop problems we use the following sign

conventions for labeling potential differences.For resistors their potential difference is negative ifFor resistors, their potential difference is negative if your loop direction and current direction are the same and it is positive if your loop direction and current direction are oppositecurrent direction are opposite.For batteries, their potential difference is positive if the direction of the loop is from the negative terminal

d h i i d i i i if h ltoward the positive and it is negative if the loop direction is from the positive terminal toward the negative.


Multiloop circuit Problem solving strategy..

Label terminals of the battery.Choose the direction of the currents you will be using and label them clearly. (Note: this selection can be done completely arbitrarily If the direction of the “real” current is opposite thearbitrarily. If the direction of the real current is opposite the direction you have selected, your answer will be negative.)Apply Kirchhoff’s Junction Rule at one or more junctions in the circuit.Choose the paths and directions for the loops that you will consider in the circuit.Apply Kirchhoff’s Loop Rule around one or more of these loops, remembering to follow the conventions we have adoptedremembering to follow the conventions we have adopted.Solve the equations algebraically for the unknowns.


A sample problem

I1Loop 1

I2I3

L 22 Loop 2


Setting up the problem

Junction Rule applicationI1 = I2 + I31 2 3

Loop 19 – I3(15 Ω) + I1(22 Ω) = 09 I3(15 Ω) + I1(22 Ω) 0

Loop 2Loop 26 + I3(15 Ω) = 0


Measuring Voltage and Currents

Ideal vs Real meters


Circuits with resistors and capacitors

We will deal with a simple series RC circuit and look at how the currents

d lt i thiand voltages in this circuit change with timetime.


Let’s consider charging the g gcapacitor first…

Using the definition of capacitance, where C = Q/ΔV andKirchhoff’s Loop RuleWe can write down an expression forWe can write down an expression for the voltage drops around this simple RC circuit in the following form….

Vbattery –I R – Q/C = 0 whereI = dQ/dt


I dQ/dt

The final form of this loop equation….

The final form of this loop equation is as follows:as follows:

Vbattery – (dQ/dt) R – Q/C = 0

This is a “linear, first order, differential equation”!! But don’t despair just yet weequation !! But don t despair just yet, we don’t need DE to solve this!!


Solving for Q(t)…..

CtQ

dttdQRVbattery =−− 0)()(

from... starting

tQCVtdQ

Cdt

battery

y

−=

)()(

as.... thisrewrite toalgebra some docan We

dttdQ

RCdt

−

=

)(...integrate.can then weand algebra ofbit last one

RCdt

CVtQtdQ

battery

−=

−)()(


RCdt

CVtQtdQ

battery

−=

−

getwesidesbothgIntegratin

)()(

RCtCV(Q(t)

t

battery −=−

becomes... thissidesboth tingexponentia

)ln

get......wesidesboth gIntegratin

CVAetQ

AeCVtQ

batteryRCt

RCt

battery

−

−

+=

=−

)( or....

)(

)- eC (VQ(t)

A

RCt

battery−

=

=

1

so.. 0 Q(t) then charge zero fromcapacitor thecharging are weif case in thisproblem.. theof conditions by the determined be oconstant t a is where


) eC (V Q(t) battery 1

RC time constant…

)eC (VQ(t) RCt

−= 1


) - eC (V Q(t) RCbattery= 1

Current flow in the system..

)C (eVdQ RCt

−) C (eV

dtRCbattery=

Note, that as the capacitor is charged the current gradually stops flowing around the circuit.


A question to ponder…

Suppose Vbattery= 20V, R = 10Ω and C= 2.0μF. Find Qf, I0 and the time

t t f thiconstant of this circuit.


∞

1040)102(20

166 −−

−

==

==∞=

CXFXVCV

) - eC (V)Q(t

b

RCbattery

0.2

1040)102(20

0 ==

==

AV

I

CXFXVCV

battery

battery

seconds1020)102(10 66

0

−− =Ω= XXRCR


Now..discharging the capacitorStart with the capacitor charged and connect the terminals together through a resistorthrough a resistor.


Using Kirchhoff’s Laws..

tQtdQ )()(from... starting

QdQ

CtQ

dttdQR =−−

)()(as.... thisrewrite toalgebra some docan We

0)()(

RCtQ

dttdQ −

=

...integrate.can then weand algebra ofbit last one

)()(

RCdt

tQtdQ −

=)()(


RCt(Q(t) −=

bthiidb thtiti

)ln

AetQ RCt

−=)(

becomes...thissidesboth tingexponentia

A

t.. then charge full fromcapacitor thegdischargin are weif case in this

problem.. theof conditions by the determined be oconstant t a is where

) C ( eV Q(t) RCt

battery−

=


Household wiring


Exam 2 Recap Chapter 24

•The nature of capacitors and how to calculate a quantity that measures their ability to store charge.•How to analyze capacitors connected in networks.networks.•How to calculate the energy stored in a capacitor.What is a dielectric and how they make•What is a dielectric and how they make

capacitors more effective.


Recap Chapter 25

•The meaning of electric current, and how charges move in a conductor.What is meant b esisti it and cond cti it of a•What is meant by resistivity and conductivity of a

substance.•How to calculate resistance of a conductor from its di i d it i ti itdimensions and its resistivity.•How an electromotive force (EMF) makes it possible for current to flow in a circuit.H t d l l ti i l i d•How to do calculations involving energy and power

in circuits.


Recap Chapter 26 •How to analyze circuits with multiple resistors in series and in parallel.•Rules that you can apply to analyzing circuits with•Rules that you can apply to analyzing circuits with more than one loop.•How to use the ammeter, voltmeter, ohmmeter, or potentiometer in a circuit.•How to analyze circuits that include both a resistor and capacitor.capacitor.•How electric power is distributed in the home.


From the formula sheet


chapter 26 october 9, 2012october 9, 2012people.physics.tamu.edu/webb/208_rcw/lectures/lecf... ·...

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