Lecture07 Capacitance; Capacitors in Series and Parallel

8/3/2019 Lecture07 Capacitance; Capacitors in Series and Parallel

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http://www.nearingzero.net(work009.jpg)


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Announcements

y Exam 1 is Tuesday, September 20, 5:00-6:15 pm.I need formal paperwork for any special needs by the endof lectures today.

y Test Preparation Homework for Exam 1 will be available onthe Physics 24 web site (under Handouts) late Friday. It will behanded out in lecture next Monday. This is next Tuesdayshomework. It will not be collected, and there will not berandom boardwork next Tuesday. (But do not ignore it!)

y Test 1 Room Assignments (next slide) are available on thePhysics 24 web site (under Course Information). Thisinformation will be included with Test Preparation Homework 1.


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Know the exam time!

Find your room ahead of time!

If at 5:00 on test day you are lost, go to 104 Physics and check the exam

room schedule, then go to the appropriate room and take the exam there.

Exam is from5:00-6:15 pm!

y Physics 24 Test Room Assignments, Fall 2011:

Instructor Sections Room

Dr. Hagen K, L 295 ToomeyDr. Hale E, G 104 PhysicsDr. Hor J, M 227 FultonDr. Parris B, H 125 Butler-Carlton (Civil)Dr. Peacher D, F G-5 HSSDr. Schmitt A, C 120 Butler-Carlton (Civil)

4:30 and 5:30 Exams 202 PhysicsSpecial Needs Testing Center


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More Announcements

yEx

am 1 special arrangements:Seven Test Center students. I have e-mailed all of you confirming yourappointment. Exam begins at the regular time (5:00) unless you havebeen asked by the test center to report at 4:25 pm, and no one will beadmitted after 5:15 pm, unless you have made other arrangements.

Three 5:30 exam students.

Other special cases: you should already have been in correspondence

with me or your recitation instructor. (one so far.)

Five 4:30 exam students.

Anybody else must let me know by the end of the 1:00 lecture todayabout special needs for the exam.


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One More Announcement

Useful for tomorrows homework: the diameter of a pennyis 1.905 cm. If you want to round that up to 2 cm, its OKwith me.


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Todays agenda:

Capacitance.You must be able to apply the equation C=Q/V.

Capacitors: parallel plate, cylindrical, spherical.You must be able to calculate the capacitance of capacitors having these geometries, andyou must be able to use the equation C=Q/V to calculate parameters of capacitors.

Circuits containing capacitors in series and parallel.You must be understand the differences between, and be able to calculate the equivalentcapacitance of, capacitors connected in series and parallel.


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Capacitors and Dielectrics

Capacitance

A capacitor is basically two parallel

conducting plates with air or insulatingmaterial in between.

V0 V1

E

L

A capacitor doesnthave to look likemetal plates. Capacitor for use in

high-performanceaudio systems.


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When a capacitor is connected to an external potential,

charges flow onto the plates and create a potential differencebetween the plates.

Capacitor platesbuild up charge.

The battery in this circuit has some voltage V. We havent discussed what

that means yet.

The symbol representing a capacitor in anelectric circuit looks like parallel plates.

Heres the symbol for a battery, or an externalpotential. +-

-

-V

+-


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If the external potential isdisconnected, charges remain on the

plates, so capacitors are good forstoring charge (and energy).

Capacitors are also very good at releasingtheir stored charge all at once. The capacitors

in your tube-type TV are so good at storingenergy that touching the two terminals at thesame time can be fatal, even though the TVmay not have been used for months.

High-voltage TV capacitors are supposed to have bleederresistors that drain the charge away after the circuit isturned off. I wouldnt bet my life on it.

Graphic from http://www.feebleminds-gifs.com/.

+

+

-

-V

conducting wires

On-line toy here.


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assortment ofcapacitors


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The magnitude of charge acquired by each plate of a capacitoris Q=CV where C is the capacitance of the capacitor.

The unit of C is the farad but most capacitors have valuesof C ranging from picofarads to microfarads (pF to QF).

micro 10-6, nano 10-9, pico 10-12 (Know for exam!)

QC

V

! C is always positive.

+Q

+

-Q

-V

CHeres this V again.In this case it is thepotential difference

provided by theexternal potential.For example, thevoltage of a battery.

V is really(V.


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Todays agenda:


Capa

citors: pa

ra

llel pla

te, cylindrica

l, spherica

l.You must be able to calculate the capacitance of capacitors having these geometries, andyou must be able to use the equation C=Q/V to calculate parameters of capacitors.



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Parallel Plate Capacitance

V0 V1

E

d

We previously calculated the electric fieldbetween two parallel charged plates:

0 0

QE .

A

W! !

I I

This is valid when the separation is smallcompared with the plate dimensions.

We also showed thatE and (V are related:

+Q-Q

A

d d

0 0V E d E dx Ed .( ! ! ! &&

"

0

0

AQ Q QC

V Ed dQd

A

I! ! ! !

(

I

This lets us calculate C fora parallel plate capacitor.


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Reminders:Q

C

V

!

Q is the magnitude of the charge on either plate.

V is actually the magnitude of the potential differencebetween the plates. V is really |(V|. Your book calls itVab.

C is always positive.


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V0 V1

E

d

+Q-Q

A

0ACd

I!

Parallel plate capacitance depends onlyon geometry.

This expression is approximate, and mustbe modified if the plates are small, or

separated by a medium other than avacuum (lecture 9).

0A

C d

OI

!

Greek letter Kappa. Fortodays lecture (and forexam 1), use Kappa=1.


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We can also calculate the capacitance of acylindrical capacitor (made of coaxialcylinders).

L

P

Coaxial Cylinder Capacitance

The next slide shows a cross-section view ofthe cylinders.


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Q

-Q

br

a

E

d"

Gaussiansurface

Q L L

C = = = bV V2k ln

a

02 LL

C = =b b2k ln lna a

Lowercase c is capacitance per unit length: 02C

c = =bL

ln

a

2kE =

r

This derivation is sometimes neededfor homework problems! (Hint: 24.10, 24.11.)

b b

b a r

a a

V = V -V = - E d = - E dr &&"

b

a

dr bV = - 2k = - 2k ln

r a


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Isolated Sphere Capacitance

An isolated sphere can be thought of as concentric sphereswith the outer sphere at an infinite distance and zero potential.

We already know the potential outside a conducting sphere:

0

Q

V .4 r! TI

The potential at the surface of a charged sphere of radius R is

0

QV

4 R!

TIso the capacitance at the surface of an isolated sphere is

0

QC 4 R.

V! ! TI


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Capacitance of Concentric Spheres

Lets calculate the capacitance of a concentric sphericalcapacitor of charge Q. Ill skip this calculation if there is no related homework assigned.

In between the spheres

2

0

QE 4 r! TI

b

2a

0 0

Q dr Q 1 1V

4 r 4 a b

( ! ! TI TI-

04Q

C1 1V

a b

TI! !

( -

You need to do this derivation if you have aproblem on spherical capacitors! (like this semester)

+Q

-Q

b

a

If there is related homework, details will be provided in lecture!


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04Q

C

1 1Va b

TI! !

( -

Let apR and bpg to get the capacitance of an isolatedsphere.

+Q

-Q

b

a

alternative calculation of capacitance of isolated sphere


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Example: calculate the capacitance of a capacitor whose platesare 20 cm x 3 cm and are separated by a 1.0 mm air gap.

d = 0.001area = 0.2 x 0.03

If you keep everything in SI (mks) units, the result is automatically in SI units.

0A

Cd

I!

128.85 10 0.2 0.03

C0.001

v v!

12C 53 10 F! v

C 53 pF!


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Example: what is the charge on each plate if the capacitor isconnected to a 12 volt* battery?

0 V

+12 V

(V= 12V

Q CV!

12Q 53 10 12! v

10Q 6.4 10 C

! v

*Remember, its the potential difference that matters.



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Example: what is the electric field between the plates?

0 V

+12 V

(V= 12V

d = 0.001

E

VE

d

(!

12V

E 0.001 m!

VE 12000 ,"up."

m

!&



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Quiz time (maybe for points, maybe just for practice!)


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Demo: Professor Tries to Avoid

Spot-Welding His Fingersto the Terminals of aCapacitor

While Demonstrating Energy Storage


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Todays agenda:


Capacitors: parallel plate, cylindrical, spherical.You must be able to calculate the capacitance of capacitors having these geometries, andyou must be able to use the equation C=Q/V to calculate parameters of capacitors.



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Capacitors in Circuits

Recall: this is the symbol representing acapacitor in an electric circuit.

And this is the symbol for a battery +-

or this

or this.


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Capacitors connected in parallel:C1

C2

C3

+ -

V

The potential difference (voltage drop) from a to b must equal V.

a b

Vab = V = voltage drop across each individual capacitor.

Vab

Circuits Containing Capacitors in Parallel

Note how I have introduced the idea that when circuit components are connected in parallel, then the voltage

drops across the components are all the same.Y

ou ma

y use this fa

ct in homework solutions.

C2

C3

+ -


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C1

C2

C3

+ -

V

a

Q = C V

Q1 = C1V

& Q2 = C2V

& Q3 = C3V

Now imagine replacing the parallelcombination of capacitors by a singleequivalent capacitor.

By equivalent, we mean stores the sametotal charge if the voltage is the same.

Ceq

+ -

V

a

Q1 + Q2 + Q3 = CeqV = Q

Q3

Q2

Q1

+ -

Q

Important!


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Q1 = C1V Q2 = C2V Q3 = C3V

Q1 + Q2 + Q3 = CeqV

Summarizing the equations on the last slide:

U

sing Q1 = C1V, etc., givesC1V + C2V + C3V = CeqV

C1 + C2 + C3 = Ceq (after dividing both sides by V)

Generalizing:

Ceq = 7Ci (capacitors in parallel)

C1

C2

C3

+ -

V

a b


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Capacitors connected in series:C1 C2

+ -

V

C3

An amount of charge +Q flows from the battery to the left plateof C1. (Of course, the charge doesnt all flow at once).

+Q -Q

An amount of charge -Q flows from the battery to the rightplate of C3. Note that +Q and Q must be the same inmagnitude but of opposite sign.

Circuits Containing Capacitors in Series


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C1 C2

+ -

V

C3

+QA

-QB

The charges +Q and Q attract equal and opposite charges tothe other plates of their respective capacitors:

-Q +Q

These equal and opposite charges came from the originallyneutral circuit regions A and B.

Because region A must be neutral, there must be a charge +Q

on the left plate of C2.

Because region B must be neutral, there must be a charge -Qon the right plate of C2.

+Q -Q


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C1 C2

+ -

V

C3

+QA

-QB

-Q +Q+Q -Q

Q = C1V1 Q = C2V2 Q = C3V3

The charges on C1, C2, and C3 are the same, and are

But we dont know V1, V2, and V3 yet.

a b

We do know that Vab = V and also Vab = V1 + V2 + V3.

V3V2V1

Vab

Note how I have introduced the idea that when circuit components are connected in series, then the voltagedrop across all the components is the sum of the voltage drops across the individual components. This isactually a consequence of the conservation of energy. You may use this fact in homework solutions.


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Collecting equations:

Q = C1V1 Q = C2V2 Q = C3V3

Vab = V = V1 + V2 + V3.

Q = CeqV

Substituting for V1, V2, and V3:1 2 3

Q Q Q V = + +C C C

Substituting for V:eq 1 2 3

Q Q Q Q= + +

C C C C

Dividing both sides by Q:eq 1 2 3

1 1 1 1= + +

C C C C

Important!


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Generalizing:

OSE: (capacitors in series)ieq i

1 1=C C


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Summary:

Series

eq i

i

C C!

same Q, Vs add

Parallel

same V, Qs add

ieq i

1 1

C C!

C1 C2 C3

C1

C2

C3


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C3

C2

C1

I dont see a series combination of capacitors, but I do see aparallel combination.

C23 = C2 + C3 = C + C = 2C

Example: determine thecapacitance of a single capacitor

that will have the same effect asthe combination shown. UseC1 = C2 = C3 = C.


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C1= CC23 = 2C

Now I see a series combination.

eq 1 23

1 1 1= +

C C C

eq

1 1 1 2 1 3

= + = + =C C 2C 2C 2C 2C

eq

2C = C

3


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Example: for the capacitor circuit shown, C1 = 3QF, C2 = 6QF,C3 = 2QF, and C4 =4QF. (a) Find the equivalent capacitance.

(b) if(V=12 V, find the potential difference across C4.

Ill work this at the blackboard.

C3

C2C1C4

(V

Homework Hint: each capacitor has associatedwith it a Q, C, and V. If you dont know what to donext, near each capacitor, write down Q= , C= ,and V= . Next to the = sign record the knownvalue or a ? if you dont know the value. As soonas you know any two of Q, C, and V, you candetermine the third. This technique often provides

visual clues about what to do next.


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You really need to know this:

Capacitors in seriesall have the same chargeadd the voltages to get the total voltage

Capacitors in parallelall have the same voltageadd the charges to get the total charge

(and it would be nice if you could explain why)


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Lecture07 Capacitance; Capacitors in Series and Parallel

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