prel12 ch27 r in series parallel

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/


Today we are going to discuss:

Chapter 28:

Section 28.4,6 Series/Parallel Resistors Section 28.7 (Example 28.29)


+ +

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

=


+ +

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

∆∆

+ +


Exa

mpl

e:

Ana

lyzi

ng a

com

plex

cir

cuit

a)Findtheequivalentresistance.b)Findthecurrentthroughandthepotentialdifferenceacrosseachoftheresistorsinthecircuit.


Real batteries


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


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/


Today we are going to discuss:

Chapter 28:

Section 28.2 Kirchhoff’s Laws Example 28.10 Analyzing a two-loop circuit


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


Kirchhoff’s Junction LawFor a junction, the law of conservation of current requires that:

1 2

3

in

out

Atanyjunctionpoint,thesumofallcurrentsenteringthejunctionmustequalthesumofallcurrentsleavingthejunction.


Kirchhoff’s Loop LawForanypaththatstartsandendsatthesamepoint:

Thesumofallthepotentialdifferencesencounteredwhilemovingaroundalooporclosedpathiszero.

Now,weneedtolearnhowtocalculatetheseΔV.Let’sstartwithabattery:


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


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


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


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


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.


Thank you

prel12 ch27 r in series parallel

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