physics for scientists and engineers ii, summer semester 2009 1 lecture 17: july 1 st 2009 physics...

Physics for Scientists and Engineers II , Summer Semester 2009

1

Lecture 17: July 1st 2009

Physics for Scientists and Engineers II


2

Example (32.1 in book): Inductance of a Solenoid

IAL

N

INL

AIL

NBA

IL

NIn

2

0B

0B

00

)3

:flux a creates B field thissolenoid theof TURN EACHThrough )2

B

:it through Icurrent a running when solenoid a inside created field magnetic The )1


3

Energy in a Magnetic Field

+ -

I

R

L

0

Iby Multiply 0

2

RIdt

dIILI

IRdt

dIL

power provided by battery power dissipated in resistor(heat)

Energy storedin the inductorper unit time(rate of energy storage)


4

Energy in a Magnetic Field

02 RIdt

dIILI

Additional energy stored in the magnetic field of the inductor per unit time (rate of energy storage).

Note: While the current increases (dI/dt > 0), amount of energy in the inductor increases.Once the maximum current is reached: dI/dt=0 and the energy in the inductor no longer increases (it has then reached it’s maximum).

2I

0 2

1dUU

:inductor theof field magnetic in the storedenergy Total

dUaddedEnergy

energy of storage of Rateunit time

addedEnergy

LIIdIL

ILdIdt

dIIL

dt

dUor


5

Energy density of a Magnetic Field

0

2

B0

220B

2

202

2

0

2

0

2u expresscan weB Using

2

1u :density Energy

2

1U :field magnetic sit'in storedenergy Total

L :Solenoid

BIn

InV

U

LI

VnALL

NA

L

N


6

Mutual Inductance

1I

1B

Coil 1 with N1 turns

Coil 2 with N2 turns

).induction"-Self" ocontrast t(in

Induction" Mutual" called is process This

2 coilin induced is emfAn changes

:1 coilin Icurrent in the change a Imagine

1

1

B

1

12212

12

12

M Inductance Mutual of Definition

1. coil respect toth wi

2 coil of inductance Mutual:M

2 coil through passingbut

1 coilby causedflux Magnetic:

I

N

)L Inductance-Self to(CompareI

N B


7

Mutual Inductance

dt

dIM

dt

dIM

MMMdt

dIM

dt

dIM

N

I

dt

dN

dt

dN

N

I

I

N

12

21

12212

211

112

2

1122

1222

2

11212

1

12212

and

shown that becan it Also, :Similarly

M:2 coilin 1 coilby induced emf

MM


8

Example: Mutual Inductance between two long wire loops exactly on top of each other.

a2

w

w)(D D

SW

Ia

awDDdr

rwr

I

Ddrrwr

IdA

rw

IdA

r

I

W

aw

a

ln11

2

11

222d

:last time showed e

00B

000B


9

Example: Mutual Inductance between two long wire loops exactly on top of each other.

loop)each for L as here same theis M :(Note

lnln1

NM

ln

loop).first theof on topexactly is loop (the

same theis loop second gh theflux throu theHowever,

lnis 1 loopgh flux throu theSo,

12

0

1

10

1

12212

10

B12

10

B

a

awD

I

Ia

awD

I

Ia

awD

Ia

awD


10

Example: Create several winding on each loop

turns.N has 2 loop and turnsN has 1 Loop :now Imagine 21

10

21

12212

110

12

lnM : and

ln :Now

Na

awDN

I

N

NIa

awD

20

12

21121

220

21

lnM : and

ln :Also

Na

awDN

I

N

NIa

awD

a

awDNNM lnMM 0

211221


11

Oscillation in an LC Circuit

closed? isswitch when thehappensWhat

).resistance no circuit, (ideal

left the toshowcircuit in the

charge) theis (Qcapacitor charged a Imagine max

+-

Lmax, QC


12

Use Kirchhoff’s loop rule

0

00C

tQ

2

2

dt

tQdLCtQ

dt

tdQ

dt

dLCtQ

dt

tdIL

+-

L C

tQtVc

dt

tdQtI

loop


13

Try this solution

equation. aldifferenti theintoit plug and

cos

sin

cos

:form thisofsolution aTry

0

2

2

2

φωt Aω(t)Q

φωtAω(t)Q

φωtA Q(t)

dt

tQdLCtQ


14

Plug it into the differential equation

LCLCω

φωt LCAωφωtA

101

0coscos

2

2

circuit. theof conditions initial by the determined areThey

)? (A, ablesother vari about theWhat


15

Initial Conditions

AAAQ

0coscos

and

0)sin(-A0(0)Q

QQ(0) and 0I(0) :Example

circuit. theof conditions initial by the determined are andA

max

max

tQtI

tQtQ

sin)(

cos

max

max


16

-3

0

time

Q(t)

I(t)

Plotting Q(t) and I(t)

tLC

QtQtItQtQ sinsin)(cos max

maxmax

maxQ

maxI

4

T

2

T4

3T T


17

Energy in the system

inductor. andcapacitor between forth andback Sloshes :Energy

tC

Q

C

tQt 2

2max

2

c cos22

1 U:Capacitorin Energy

tILtILt 22max

2L sin

2

1

2

1 U:Inductorin Energy

C

Qtt

C

Q

tLC

QLt

C

Q

tILtC

Qtt

2sincos

2

sin2

1cos

2

sin2

1cos

2UUt U:Energy Total

2max22

2max

22max2

2max

22max

22max

Lc

constant2

U2max C

Q


18

Energy in the system

Uc(t)

0

0

timeUL(t)

0

0

time2

T

4

T

4

3T T

C

Q

2

2max

C

QLI

22

1 2max2

max


19

RLC Circuits (more realistic)

behave?circuit thedoes How

b"." switch to flip Then,

time).long afor it there leave (and

a"" switch to flipFirst

:Imagine

L C

R

a

b

".freely swing" circuit to theofpart

right theleaves andbattery theisolates b"" switch to theFlipping

value.intitial some tocapacitor charges a"" switch to Flipping


20

Kirchhoff’s loop

C0)Q(t :capacitoron Charge

:b"" toswitchingafter y Immediatel

L CtQ )0(

R

a

b

0)()(

)(0)(

:rule loop sKirchhoff'

2

2

RCdt

tdQ

dt

tQdLCtQIR

dt

dIL

C

tQ

dt

tdQtI

)()(


21

Kirchhoff’s loop

0)()()(

0)()(

)(

2

2

2

2

tQRCdt

tdQ

dt

tQdLC

RCdt

tdQ

dt

tQdLCtQ

tAt

tABt

tAt

t

dddd

ddd

ddd

d

cosesineAB

sinecoseAB(t)Q

sinecoseAB(t)Q

coseAQ(t)

:n)oscillatio damped (asolution Try this

Bt-2Bt-

Bt-Bt-2

Bt-Bt-

Bt-


22

Plug Trial Solution into Diff. Equation

terms)(cosine01B

terms)(sine0B2

0cossincosB

cossin2cosBC

0coseAsinecoseAB

cosesine2coseABC

22

22

Bt-Bt-Bt-

-Bt2-Bt-Bt2

RCBLCLC

RL

tttRC

ttBtL

ttAtRC

tAtABtL

d

dddd

ddddd

dddd

ddddd


23

Plug Trial Solution into Diff. Equation

terms)(cosine01B

terms)(sine0B2

22

RCBLCLC

RL

d

2

2

2

2

2

2

2

22

2

1

4

1

42

1

11

L

R

LC

L

R

LCL

R

L

R

LC

BL

RB

LCLC

LCBRCBd

L

RB

2

tdcoseAQ(t) 2LRt-


24

Initial Condition

max)0( QtQ

A 0coseAQ -0max

tdcoseQQ(t) 2LRt-

max

2

2

1 where

L

R

LCd

physics for scientists and engineers ii, summer semester 2009 1 lecture 17: july 1 st 2009 physics...

Documents

summer semester

solution slide

system slide

realistic slide

solenoid slide

mutual inductance slide

kirchhoffs loop slide

magnetic field slide