Lecture09 Energy Stored in Capacitors and Electric Fields; Dielectrics

8/3/2019 Lecture09 Energy Stored in Capacitors and Electric Fields; Dielectrics

1/29

http://www.nearingzero.net(nz045.jpg)


2/29

Announcements

y Physics 24 spreadsheets for all sections, with Exam 1 scores, arenow posted on the Physics 24 web site. You need your PIN to findyour grade.

y Physics 24 Exam 1 will be returned in recitation Thursday. Whenyou get the exam back, please check that points were addedcorrectly. Review the course handbook and be sure to followproper procedures before requesting a regrade. Get yourregrade requests in on time! (They are due by next Tuesdaysrecitation.)

y Preliminary exam average is about 76.3%. Quite reasonable!Scores ranged from a low of 41 to a high of 200 (seven students).


3/29

Todays agenda:

Energy Storage in Capacitors.You must be able to calculate the energy stored in a capacitor, and apply the energystorage equations to situations where capacitor configurations are altered.

Dielectrics.You must understand why dielectrics are used, and be able include dielectric constants incapacitor calculations.


4/29

Energy Storage in Capacitors

Lets calculate how much work it takes to charge a capacitor.

The work required for an external force to move a charge dqthrough a potential difference (V is dW = dq (V.

(V+ -

+q -q

+

dq

From Q=C(V ( (V = q/C):

qdW dq

C!

q is the amount of charge onthe capacitor at the time thecharge dq is being moved.

We start with zero charge on the capacitor,and end up with Q, so

Q2 2

Q Q

0 00

q q QW dW dq .

C 2C 2C! ! ! !


5/29

The work required to charge the capacitor is the amount ofenergy you get back when you discharge the capacitor

(because the electric force is conservative).

Thus, the work required to charge the capacitor is equal to thepotential energy stored in the capacitor.

2QU .

2C!

Because C, Q, and V are related through Q=CV, there are three

equivalent ways to write the potential energy.

2 2Q CV QVU .

2C 2 2! ! !


6/29

2 2Q CV QV

U .

2C 2 2

! ! !

All three equations are valid; use the one most convenient forthe problem at hand.

It is no accident that we use the symbol U for the energystored in a capacitor. It is just another version of electricalpotential energy. You can use it in your energy conservationequations just like any other form of potential energy!


7/29

Example: a camera flash unit stores energy in a 150 QFcapacitor at 200 V. How much electric energy can be stored?

2CVU

2!

6 2

150 10 200U

2

v!

U 3 J!

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


8/29

Example: compare the amount of energy stored in a capacitorwith the amount of energy stored in a battery.

A12 V car battery rated at 100 ampere-hours stores 3.6x105 Cof charge and can deliver at least 4.3x106joules of energy(well learn how to calculate that later in the course).

If a battery stores so much more energy, why use capacitors?

Application #1: short pulse magnets at the National MagnetLaboratory, plus a little movie of a short pulse magnet at work.

106joules of energy are stored at high voltage in capacitorbanks, and released during a period of a few milliseconds. Theenormous current produces incredibly high magnetic fields.

A100 QF capacitor that stores 3.6x105 C at 12 V stores an

amount of energy U=CV2

/2=7.2x10-3

joules. If you want your capacitorto store lots of energy, storeit at a high voltage.


9/29

Application #2: quarter shrinker.

Application #3: can crusher.


10/29

Energy Stored in Electric Fields

(V+ -

+Q -Q

E

21U C V

2! (

d

area A

2

0A1

U Ed2 d

I

!

201

U Ad E2

! I

Energy is stored in thecapacitor:

The volume of the capacitor is Volume=Ad


11/29

Energy stored per unit volume (u):

(V+ -

+Q -Q

E

d

area A

2

02

0

1

Ad E 12u EAd 2

I! ! I

The energy is stored in the electric

field!

Weve gone from the concrete(electric charges experience forces)

to the abstract (electric chargescreate electric fields)

to an application of the abstraction(electric field contains energy).


12/29

2

0

1u E

2! I

(V+ -

+Q -Q

E

f

area A

This is not a new kind of energy. Its theelectric potential energy resulting from thecoulomb force between charged particles.

Or you can think of it as the electric energydue to the field created by the charges.Same thing.

The energy in electromagnetic phenomena is the same as mechanicalenergy. The only question is, Where does it reside?In the old theories, itresides in electrified bodies. In our theory, it resides in the electromagnetic

field, in the space surrounding the electrified bodies.James Maxwell

This is on your OSE sheet.Do not use until later!

(Unless you really know whatyou are doing.)


13/29

Todays agenda:

Energy Storage in Capacitors.You must be able to calculate the energy stored in a capacitor, and apply the energystorage equations to situations where capacitor configurations are altered.

Dielectrics.You must understand why dielectrics are used, and be able include dielectric constants incapacitor calculations.


14/29

If an insulating sheet (dielectric) isplaced between the plates of a

capacitor, the capacitance increasesby a factor O, which depends on thematerial in the sheet. O is the

dielectric constant of the material.

dielectric

In general, C = OI0A / d. O is 1for a vacuum, and } 1 for air.(You can also define I = OI0and write C = I A / d).

Dielectrics

OI A

C = .d


15/29

The dielectric is the thin insulating sheetin between the plates of a capacitor.

dielectric

Any reasons to use a dielectric in a capacitor?

yLets you apply higher voltages (so more charge).

yLets you place the plates closer together (make dsmaller).

yIncreases the value of C because O>1.Q= CV

OI A

C =d

yMakes your life as a physics student more complicated.

Gives you a bigger kick whenyou discharge the capacitor

through your tongue!






16/29

dielectric

This is equivalent to two capacitors in parallel. Each of the

two has half the plate area. The two share the totalcharge, and have the same potential difference

Homework hint: what if the

dielectric fills only half thespace between the plates?

C1

C2Q2

Q1

CQ


17/29

Homework hint: if you charge a capacitor and then remove

the battery and manipulate the capacitor, Q must stay thesame but C, V, and U may change. (What about E?)

Homework hint: if you charge a capacitor, keep the batteryconnected, and manipulate the capacitor, V must stay thesame but C, Q, and U may change. (What about E?)

Homework hint: if exactly two capacitors are connectedsuch that they have the same voltage across them, they areprobably in parallel (but check the circuit diagram).


18/29

Homework hint: if you charge two capacitors, then remove

the battery and reconnect the capacitors with oppositely-charged plates connected together

draw a circuit diagram before and after, and useconservation of charge to determine the total charge on

each plate before and after.


19/29

Example: a parallel plate capacitor has an area of 10 cm2 andplate separation 5 mm. 300 V is applied between its plates. Ifneoprene is inserted between its plates, how much chargedoes the capacitor hold.

(V=300 Vd=5 mm

A=10 cm2

O=6.7

-12 -4-3

6.7 8.8510 1010C =

510

Q=CV

v-11

C =1.19 10 F

v ! v-11 -9Q= 1.19 10 300 3.56 10 C =3.56 nC

OI A

C =d


20/29

Example: how much charge would the capacitor on theprevious slide hold if the dielectric were air?

The calculation is thesame, except replace 6.7by 1.

Or just divide the charge on theprevious page by 6.7 to get.

Q=0.53 nC (V=300 Vd=5 mm

A=10 cm2

O=1


21/29

Visithowstuffworks to read about capacitors and learn theiradvantages/disadvantages compared to batteries!

A capacitor connected as shown

acquires a charge Q.V

While the capacitor is still connectedto the battery, a dielectric material isinserted.

Will Q increase, decrease, or stay the same?

Why?

V

V=0Conceptual Example


22/29

Example: find the energy stored in the capacitor in slide 20.

(V=300

V

d=5 mm

A=10 cm2

O=6.7

v-11C =1.19 10 F

12

(2

U= C V

1

2v

2-11U= 1.19 10 300

v-7U=5.36 10 J


23/29

Example: the battery is now disconnected. What are thecharge, capacitance, and energy stored in the capacitor?

(V=300

V

d=5 mm

A=10 cm2

O=6.7

The charge and capacitance areunchanged, so the voltage dropand energy stored are unchanged.

Q =3.56 nC

v-11C =1.19 10 F

v-7

U=5.36 10 J


24/29

Example: the dielectric is removed without changing the plateseparation. What are the capacitance, charge, potentialdifference, and energy stored in the capacitor?

(V=300

V

d=5 mm

A=10 cm2

O=6.7 -12 -4-3

8.8510 1010C =

510

v-12C=1.78 10 F

I A

C =d

(V=?d=5 mm


25/29

(V=?d=5 mm

A=10 cm2

Q =3.56 nC


The charge remains unchanged,because there is nowhere for itto go.


26/29

(V=?d=5 mm

A=10 cm2

v(

v

-9

-12

3.56 10QV = =

C 1.78 10

(V = 2020 V


Knowing C and Q we cancalculate the new potentialdifference.


27/29

(V=2020

V

d=5 mm

A=10 cm2

12

(2

U= C V

12

v2-12U= 1.78 10 2020

v -6U=3.63 10 J



28/29

v-7

beforeU =5.36 10 J

v -6afterU =3.63 10 J

after

before

U=6.7

U

Huh?? The energy stored increases by a factor ofO??

Sure. It took work to remove the dielectric. The stored energy

increased by the amount of work done.

( externalU= W


29/29

A toy to play with

http://phet.colorado.edu/en/simulation/capacitor-lab

(You might even learn something.)

Lecture09 Energy Stored in Capacitors and Electric Fields; Dielectrics

Documents

Transcript of Lecture09 Energy Stored in Capacitors and Electric Fields; Dielectrics