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PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Lecture 12
Chapter 28
Resistors in Series and Parallel
Physics II
Finally! Spring Break! I can forget about
him for a week
Course website:https://sites.uml.edu/andriy-danylov/teaching/physics-ii/
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Today we are going to discuss:
Chapter 28:
Section 28.4,6 Series/Parallel Resistors Section 28.7 (Example 28.29)
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
+ +
Resistors in ParallelConsider three resistors connected in parallel.
I
Real
circ
uit
Equi
vale
nt c
ircui
t
ΔV
Resistors in parallel have the same potential difference, ΔV
I + +
;
We have replaced 3 resistors with an “equivalent” resistor.
+ +
Conservation of current
Req is inserted without changing the operation of the circuit, so I and ΔV are same as in the real circuit
Equivalent resistance of resistors in parallel.
=
I1
I2
I3
Ohm’s law;
ΔV
=
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
+ +
Resistors in SeriesConsider three resistors connected in series.
Rea
l cir
cuit
Equ
ival
ent c
ircu
it
ΔV
+ +
Ohm’s law ∆
Req is inserted without changing the operation of the circuit, so I and ΔV are same as in the real circuit
Equivalent resistance of resistors in series.
ΔV
ΔV1 ΔV2 ΔV3
∆∆
+ +
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Exa
mpl
e:
Ana
lyzi
ng a
com
plex
cir
cuit
a)Findtheequivalentresistance.b)Findthecurrentthroughandthepotentialdifferenceacrosseachoftheresistorsinthecircuit.
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Real batteries
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Give me a break! I do it as fast as I
can!
Real Batteries. Internal resistance
Todriveacurrentinacircuitweneeda“chargepump”,adevicethatbydoingworkonthechargecarriersmaintainsapotentialdifference.Let’slookatagravitationalanalogofabattery:
Apersondoesworktomaintainasteadyflowofballsthrough“thecircuit”.However,thisguycannotmoveballsinstantaneously.Ittakestime.Sothereisanaturalhindrancetoacompletelyfreeflow.Todescribethishindrancewecanintroducetheinternalresistance,r.Itisinsideabatteryanditcannotbeseparatedfromthebattery.
Pot. difference of a battery without an internal resistance is called an electromotive force.(EMF, ε)
∆Terminal voltage
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Lecture 12
Chapter 28
Kirchhoff’s Laws
Physics II
Finally! Spring Break! I can forget about
him for a week
Course website:https://sites.uml.edu/andriy-danylov/teaching/physics-ii/
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Today we are going to discuss:
Chapter 28:
Section 28.2 Kirchhoff’s Laws Example 28.10 Analyzing a two-loop circuit
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Kirchhoff’s RulesIdea! Some circuits are too complicated to analyze
(none of the elements are in series/parallel)
Kirchhoff’s rules are very helpful.
Toanalyzeacircuitmeanstofind:1. ΔVacrosseachcomponent2. Thecurrentineachcomponent
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Kirchhoff’s Junction LawFor a junction, the law of conservation of current requires that:
1 2
3
in
out
Atanyjunctionpoint,thesumofallcurrentsenteringthejunctionmustequalthesumofallcurrentsleavingthejunction.
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Kirchhoff’s Loop LawForanypaththatstartsandendsatthesamepoint:
Thesumofallthepotentialdifferencesencounteredwhilemovingaroundalooporclosedpathiszero.
Now,weneedtolearnhowtocalculatetheseΔV.Let’sstartwithabattery:
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
ΔV across a battery
Travel direction
Travel direction
Higher VLower V
Final pointInitial point according to a travel direction
Higher V Lower V
Final pointInitial point
Δ
Δ
according to a property of a battery
For a battery, the potential difference is positive if your chosen loop direction is from the negative terminal toward the positive terminal
The potential difference is negative if the loop direction is from the positive terminal toward the negative terminal
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
ΔV’s across resistors
Current direction
Travel direction
Current direction
Travel direction
+ _
Higher V Lower V
Initial point according to a travel direction
Final point
(Because I flows from higher V to lower V)
_ +
Δ
Δ
For a resistor, apply Ohm’s law; the potential difference is negative (a decrease) if your chosen loop direction is the same as the chosen current direction through that resistor
For a resistor, apply Ohm’s law; the potential difference is positive (an increase) if your chosen loop direction is opposite to the chosen current direction through that resistor
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Let’s take a look at how the junction rule and loop rule help us solve for the unknown values in multi-loop circuits.
Example Multi-Loop Circuit
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
No junction points
Loop rule
1) Assume CW direction of current(If our assumption turns out to be wrong, the current will be negative)
=
=2) Choose a travel direction (say, CW) and a start point
Travel direction=
+ ‐
+‐
Now we can find pot. differences across each resistor
Example Example 28.1. Analyze the circuit
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Tactics: Using Kirchhoff’s Rules1. Label the current in each separate branch of the given circuit with a different subscript, suchas Each current refers to a segment between two junctions. Choose the direction ofeach current, using an arrow. The direction can be chosen arbitrarily: if the current is actually inthe opposite direction, it will come out with a minus sign in the solution.
1 2 3, , I I I
2. Identify the unknowns. You will need as many independent equations as there areunknowns. You may write down more equations than this, but you will find that some of theequations will be redundant (that is, not be independent in the sense of providing newinformation). You may use for each resistor, which sometimes will reduce the number ofunknown
3. Apply Kirchhoff’s junction rule at one or more junctions.
3. Apply Kirchhoff’s loop rule for one or more loops: follow each loop in one direction only. Pay careful attention to subscripts, and to signs:(a) For a resistor, apply Ohm’s law; the potential difference is negative (a decrease) if your chosen loop direction is the same as the chosen current direction through that resistor; the potential difference is positive (an increase) if your chosen loop direction is opposite to the chosen current direction.(b) For a battery, the potential difference is positive if your chosen loop direction is from the negative terminal toward the positive terminal; the potential difference is negative if the loop direction is from the positive terminal toward the negative terminal.
4.Solve the equations algebraically for the unknowns.
PHYS.1440Lecture12 A.DanylovDepartmentofPhysicsandAppliedPhysics
Thank you