comments: yours and ours - university of illinois … yours and ours 1. please respect your fellow...

Comments Yours and Ours

1 Please respect your fellow students ndash

2 Quite a bit of homework this week

3 Donrsquot panicDonrsquot be intimidated by integrals Physics 212 Lecture 2 Slide 1

ldquoI find the most difficult part to determine the direction of electric field I need to calculate the magnitude as well as the angle for net magnitude and finally analysis and make the decisionrdquo ldquoi started to get confused at electric dipoles give the positive and negative charges of same magnitude how do you know what direction the field points in also the infinite line stuff made 0 senserdquo ldquoSo what that point charge is 1r^2 and infinite line is 1r Does that dimensional dependence matterrdquo ldquoI am very confused on the dipole charges and the infinite lines of charge Could you explain more about thoserdquo ldquoId like to spend more time with setting up integrals for continuous distributionsrdquo ldquoThe integral of the continuous line charges is a bit abstract for me I dont quite

understand how to add up for the Y componentrdquo ldquoI didnt get the part in superposition of vectors when he talks about dimensions and how its reflected in the formulardquo

Electricity amp Magnetism

Lecture 2

Todayrsquos Concepts A) The Electric Field

B) Continuous Charge Distributions

Electricity amp Magnetism Lecture 2 Slide 2

Pretty cool stuff man but like what does it all meannnnn

05

If there are more than two charges present the total force on any given charge is just the vector sum of the forces due to each of the other charges

+q1 -gt -q1 direction reversed

Coulombrsquos Law (from last time)

F21

F31

F41

F1

q1

q2

q3

q4

F21

F31

F41

F1

F21

F31

F41

F1

F21

F31

F41

F1

q1

q2

q3

q4

142

14

41132

13

31122

12

21 ˆˆˆ rr

qkqr

r

qkqr

r

qkq++


MATH

1F

142

14

4132

13

3122

12

2

1

1 ˆˆˆ rr

kqr

r

kqr

r

kq

q

F++

E

09

The electric field E at a point in space is simply the force per unit charge at that point Electric field due to a point charged particle

Superposition

E2

E3

E4

E

Field points toward negative and Away from positive charges

ldquoWhat is the essence of an electric field ldquo

Electric Field

q4

q2

q3


q

FE

rr

QkE ˆ

2

i

i

i

i rr

QkE ˆ

2

12

Two equal but opposite charges are placed on the x axis The positive charge is placed to the left of the origin and the negative charge is placed to the right as shown in the figure above

ldquoI cant figure out the field directions at allrdquo

CheckPoint


A

B x

y

+Q -Q

What is direction at point A

a) Up b) down c) Left d) Right e) zero

What is direction at point B


15

E

E

CheckPoint

+Q

+Q

+Q

-Q

In which of the two cases shown below is the magnitude of the electric field at the point labeled A the largest (Select C if you think they are equal)

A

A

Case A Case B


ldquoThe charges are equal distance and since they are perpendicular they dont cancel outrdquo

ldquoThe magnitudes combine together instead of canceling out like they do in case Ardquo

in case B the charges will cancel each other

18

A B Equal

Two charges q1 and q2 are fixed at points (-a0) and (a0) as shown Together they produce an electric field at point (0d) which is directed along the negative y-axis

x

y

q1 q2

(-a0) (a0)

(0d)

Which of the following statements is true A) Both charges are negative B) Both charges are positive C) The charges are opposite D) There is not enough information to tell how the charges are

related

Two Charges

E

Electricity amp Magnetism Lecture 2 Slide 8 20

_ _ + +

_ +


A B C D

ldquo(A LEFT) Since the radius is squared the distance between the 2Q charge makes its pull smaller than that of the 1Q chargeldquo

ldquo(B) Force from Q charge is greater than 2Q because kQqr^2gt2kQq4r^2 =gt 1gt12 and test charge is repelledrdquo

ldquo(C Still) the two electric field are equal in magnitude but opposite in directionrdquo

CheckPoint



Example

Calculate E at point P

ldquoMore examples of electric fields of different charge distributions would be nice (for example charges arranged in a circle or square)rdquo

d

P

d

A) B) C) D) Need to know d

Need to know d amp q

E)

What is the direction of the electric field at point P the unoccupied corner of the square

0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL

Summation becomes an integral (be careful with vector nature)

ldquoI would like to go over the process of integration with the infinite line of charge example again in class if possiblerdquo

WHAT DOES THIS MEAN

Integrate over all charges (dq)

r is vector from dq to the point at which E is defined

r

dE

Continuous Charge Distributions

Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density

Linear (l QL) Coulombsmeter

Surface (s QA) Coulombsmeter2

Volume (r QV) Coulombsmeter3

What has more net charge

A) A sphere w radius 4 meters and volume charge density r = 2 Cm3

B) A sphere w radius 4 meters and surface charge density s = 2 Cm2

C) Both A) and B) have the same net charge

ldquoWhat exactly is Lambda QL Charge densityrdquo

Some Geometry

24 RAsphere

3

34 RVsphere LRVcylinder

2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint


A) (EAltEB) ldquoAs the electric field by both lines

have equal magnitude on A but opposite

directions they get cancelled out hence electric

field at A=0 whereas that is not the case at B

and electric field at Bgt0rdquo

C) (EAgtEB) ldquoE_A will be grater becuase it has

two lines of charge acting on it from equl

distance were E_B would be less because it is

fruther from the lower line of chargerdquo

37

ldquoHow is the integration of dE over L worked out step by steprdquo

Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl

Charge is uniformly distributed along the x-axis from the origin to x a The charge density is l Cm What is the x-component of the electric field at point P (xy) (ah)

x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)

C) A and B are both OK

xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l

Putting it all together

+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem


Please try to solve this problem on your

own Only use help if you get stuck and

have thought hard about it for at least

10 minutes

Electricity amp Magnetism

Lecture 2

Todayrsquos Concepts A) The Electric Field

B) Continuous Charge Distributions


Pretty cool stuff man but like what does it all meannnnn

05




F21

F31

F41

F1

q1

q2

q3

q4

F21

F31

F41

F1

F21

F31

F41

F1

F21

F31

F41

F1

q1

q2

q3

q4

142

14

41132

13

31122

12

21 ˆˆˆ rr

qkqr

r

qkqr

r

qkq++


MATH

1F

142

14

4132

13

3122

12

2

1

1 ˆˆˆ rr

kqr

r

kqr

r

kq

q

F++

E

09


Superposition

E2

E3

E4

E



Electric Field

q4

q2

q3


q

FE

rr

QkE ˆ

2

i

i

i

i rr

QkE ˆ

2

12



CheckPoint


A

B x

y

+Q -Q





15

E

E

CheckPoint

+Q

+Q

+Q

-Q


A

A

Case A Case B





18

A B Equal


x

y

q1 q2

(-a0) (a0)

(0d)


related

Two Charges

E


_ _ + +

_ +


A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes




F21

F31

F41

F1

q1

q2

q3

q4

F21

F31

F41

F1

F21

F31

F41

F1

F21

F31

F41

F1

q1

q2

q3

q4

142

14

41132

13

31122

12

21 ˆˆˆ rr

qkqr

r

qkqr

r

qkq++


MATH

1F

142

14

4132

13

3122

12

2

1

1 ˆˆˆ rr

kqr

r

kqr

r

kq

q

F++

E

09


Superposition

E2

E3

E4

E



Electric Field

q4

q2

q3


q

FE

rr

QkE ˆ

2

i

i

i

i rr

QkE ˆ

2

12



CheckPoint


A

B x

y

+Q -Q





15

E

E

CheckPoint

+Q

+Q

+Q

-Q


A

A

Case A Case B





18

A B Equal


x

y

q1 q2

(-a0) (a0)

(0d)


related

Two Charges

E


_ _ + +

_ +


A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes


Superposition

E2

E3

E4

E



Electric Field

q4

q2

q3


q

FE

rr

QkE ˆ

2

i

i

i

i rr

QkE ˆ

2

12



CheckPoint


A

B x

y

+Q -Q





15

E

E

CheckPoint

+Q

+Q

+Q

-Q


A

A

Case A Case B





18

A B Equal


x

y

q1 q2

(-a0) (a0)

(0d)


related

Two Charges

E


_ _ + +

_ +


A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes



CheckPoint


A

B x

y

+Q -Q





15

E

E

CheckPoint

+Q

+Q

+Q

-Q


A

A

Case A Case B





18

A B Equal


x

y

q1 q2

(-a0) (a0)

(0d)


related

Two Charges

E


_ _ + +

_ +


A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

E

E

CheckPoint

+Q

+Q

+Q

-Q


A

A

Case A Case B





18

A B Equal


x

y

q1 q2

(-a0) (a0)

(0d)


related

Two Charges

E


_ _ + +

_ +


A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes


x

y

q1 q2

(-a0) (a0)

(0d)


related

Two Charges

E


_ _ + +

_ +


A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

_ _ + +

_ +


A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

A B C D




CheckPoint



Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes


Example



d

P

d



E)


0E

( )

-

4cos

222

d

q

d

qkEx

( )

-

4sin

222

d

q

d

qkEy

-q +q

+q

i

i

i

i rr

QkE ˆ

2

28

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

l QL



WHAT DOES THIS MEAN



r

dE


Linear Example

charges

pt for E

dq l dx


i

i

i

i rr

QkE ˆ

2

r

r

dqkE ˆ

2

30

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

Charge Density









Some Geometry

24 RAsphere

3


2

RLAcylinder 2

3

34 RVQA rr

24 RAQB ss

RR

R

Q

Q

B

A

s

r

s

r

3

1

4 2

3

34


3

44

2

2

3

1

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

10)

CheckPoint











37


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes


Calculation

2x

dxA) B) C) D) E)

What is 2r

dq

22 ha

dx

+22 ha

dx

+

l22)( hxa

dx

+-

l2x

dxl


x

y

a

h

P

x

r

dq l dx


We know

rr

dqkE ˆ

2

40

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

Calculation

We know

222 )( hxa

dx

r

dq

+-

l

xx dEE

What is

A) B) C) D)

xdE

1cosdE2cosdE

1sindE 2sindE


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2 We want

42

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

We know

cos2 DEPENDS ON x

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l

What is

A) B)


xE

-+

a

axh

dxk

0

222)(

cosl +-

a

hxa

dxk

0

22

2

)(

cosl


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

45

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

We know

Calculation

2cosdEEx222 )( hxa

dx

r

dq

+-

l


x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2

22 ha

x

+

What is

A) B) C) 22 ha

a

+ 22)( hxa

a

+- D)

22 ha

a

+22)( hxa

a

+-22)( hxa

xa

+-

-

2cos

47

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

We know

xdE

Calculation

222 )( hxa

dx

r

dq

+-

l 2cosdEEx

222

)(cos

hxa

xa

+-

-

( ) 23220 )(

)(hxa

xadxkPE

a

x

+-

- l


+-

221)(

ah

h

h

kPEx

l

x a

P

x

r

1 2

2

dq l dx

h

xdE

dE

y


rr

dqkE ˆ

2


49

Homework Problem





10 minutes

Homework Problem





10 minutes

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