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Direct-Current Circuits Direct Current Circuits Chapter 26 October 9, 2012 October 9, 2012 October 9, 2012 Physics 208 1
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
October 9, 2012 Physics 208 2
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.
October 9, 2012 Physics 208 3
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…
October 9, 2012 Physics 208 4
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…
October 9, 2012 Physics 208 5
Combination circuits..
October 9, 2012 Physics 208 6
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
October 9, 2012 Physics 208 7
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
October 9, 2012 Physics 208 8
Rule # 2…the Loop RuleThe sum of the changes in potential around ANY Closed path in a circuit must be zeromust be zero
October 9, 2012 Physics 208 9
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.
October 9, 2012 Physics 208 10
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.
October 9, 2012 Physics 208 11
A sample problem
I1Loop 1
I2I3
L 22 Loop 2
October 9, 2012 Physics 208 12
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
October 9, 2012 Physics 208 13
Measuring Voltage and Currents
Ideal vs Real meters
October 9, 2012 Physics 208 14
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.
October 9, 2012 Physics 208 15
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
October 9, 2012 Physics 208 16
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!!
October 9, 2012 Physics 208 17
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
−=
−)()(
October 9, 2012 Physics 208 18
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
October 9, 2012 Physics 208 19
) eC (V Q(t) battery 1
RC time constant…
)eC (VQ(t) RCt
−= 1
October 9, 2012 Physics 208 20
) - 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.
October 9, 2012 Physics 208 21
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.
October 9, 2012 Physics 208 22
∞
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
October 9, 2012 Physics 208 23
Now..discharging the capacitorStart with the capacitor charged and connect the terminals together through a resistorthrough a resistor.
October 9, 2012 Physics 208 24
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 −
=)()(
October 9, 2012 Physics 208 25
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−
=
October 9, 2012 Physics 208 26
Household wiring
October 9, 2012 Physics 208 27
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.
October 9, 2012 Physics 208 28
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.
October 9, 2012 Physics 208 29
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.
October 9, 2012 Physics 208 30
From the formula sheet
October 9, 2012 Physics 208 31
