Transcript phys1444-lec9

PHYS 1444 – Section 004
Lecture #14
Thursday July 5, 2012
Ian Howley
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Chapter 26
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Kirchhoff’s Rules
RC Circuit
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Kirchhoff’s Rules
• Some circuits are very complicated
to analyze using the simple rules
for combining resistors
– G. R. Kirchhoff devised two rules to
deal with complicated circuits.
• Kirchhoff’s rules are based on conservation of charge
and energy
– Kirchhoff’s 1st rule: Junction rule, charge conservation.
• At any junction point, the sum of all currents entering the junction
must equal to the sum of all currents leaving the junction.
• In other words, charge is conserved
• At junction a in the figure, I3 goes into the junction while I1 and I2
leaves: I3 = I1+ I2
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Kirchhoff’s 2nd Rule
• Kirchoff’s 2nd rule: Loop rule, uses
conservation of energy.
– The sum of the changes in potential around
any closed path of a circuit must be zero.
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Kirchhoff’s
nd
2
Rule
• The current in the circuit in the figure is
I =12/690 =0.017A.
– Point e is the highest potential point while point d
is the lowest potential.
– When the test charge starts at e and returns to e,
the total potential change is 0.
– Between point e and a, no potential change since
there is no source of potential nor any resistance.
– Between a and b, there is a 400W resistance,
causing IR=0.017*400 =6.8V drop.
– Between b and c, there is a 290W resistance,
causing IR=0.017*290 =5.2V drop.
– Since these are voltage drops, we use negative
sign for these, -6.8V and -5.2V.
– No change between c and d while from d to e
there is +12V change.
– Thus the total change of the voltage through the
loop is: -6.8V-5.2V+12V=0V.
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Using Kirchhoff’s Rules
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Determine the flow of currents at the junctions.
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It does not matter which direction you assign a current.
If the value of the current after completing the calculations are
negative, it means you chose the wrong direction
Write down the current equation based on Kirchhoff’s 1st
rule at various junctions.
Choose closed loops in the circuit
Write down the potential in each interval of the junctions,
keeping the sign properly.
Write down the potential equations for each loop.
Solve the equations for unknowns.
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Example 26 – 8
Use Kirchhoff’s rules. Calculate
the currents I1, I2 and I3.
The directions of the current through the circuit is not
known a priori but since the current tends to move away
from the positive terminal of a battery, we arbitrarily
choose the direction of the currents as shown.
We have three unknowns so we need three equations.
I 3  I1  I 2
Using Kirchhoff’s junction rule at point a, we obtain
This is the same for junction d as well, so it gives no additional information.
Now apply K’s second rule to the top loop ahdcba
Vah   I1 30
Vhd  0
Vdc  45
Vcb   I 3
Vba  40I 3
The total voltage change in loop ahdcba is.
Vahdcba  30 I1  45  I3  40 I 3  45  30 I1  41I3  0
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Example 26 – 8, cnt’d
Now the 2nd rule on the bottom loop agfedcba.
Vag  0 Vgf  80 V fe   I 2 Ved   I 2 20
Vdc  45
Vcb   I 3
Vba 40I 3
The total voltage change in loop agfedcba Vagfedcba  21I 2  125  41I 3  0
So the three equations are
I 3  I1  I 2
45  30 I1  41I 3  0
125  21I 2  41I 3  0
We can obtain the three current by solving these equations for I1, I2 and I3.
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RC Circuits
• Circuits containing both resistors and capacitors
– RC circuits are used commonly in everyday life
• Control windshield wiper
• Timing of traffic light from red to green
• Camera flashes and heart pacemakers
• What does an RC circuit look like?
– There should be a source of emf, capacitors and resistors
• What happens when the switch S is closed?
– Current immediately starts flowing through the circuit.
– Electrons flow out of negative terminal of the emf source, through the resistor R and
accumulate on the upper plate of the capacitor
– The electrons from the bottom plate of the capacitor will flow into the positive
terminal of the battery, leaving only positive charge on the bottom plate
– As the charge accumulates on the capacitor, the potential difference across it
increases
– The current reduces gradually to zero -- at that point the voltage across the
capacitor is the same as that of the emf
– The charge on the capacitor increases until it reaches to its maximum CE.
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RC Circuits
• What does all this look like graphically?
– Charge on the capacitor and current as a function of time
– From energy conservation (Kirchhoff’s 2nd rule), the emf E be equal to
the voltage drop across the capacitor and the resistor
• E=IR+Q/C
• R includes all resistance in the circuit, including the internal
resistance of the battery, I is the current in the circuit at any instant,
and Q is the charge of the capacitor at that same instant
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RC Circuits
• What are the charge and voltage at any time?
– Charge on the capacitor and current as a function of time
– Example 26-11 on board
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