General Physics II, Lec 3, By/ T.A. Eleyan 1 Lecture 3 The Electric Field

General Physics II, Lec 3, By/ T.A. Eleyan

1

Lecture 3

The Electric Field


2

Electric Field

Suggests the notion of electrical field (first introduced by Michael Faraday (1791-1867).

An electric field is said to exist in a region of space surrounding a charged object.

If another charged object enters a region where an electrical field is present, it will be subject to an electrical force.


3

Consider a small charge q0 near a larger charge Q.We define the electric field E at the location of the small test charge as a ratio of the electric force F acting on it and the test charge q0

This is the field produced by the charge Q, not by the charge q0

0

/F

E N Cq

Electric Field & Electric Force


4

Electric Field Direction The direction of E at a point is the direction of the electric force that would be exerted on a

small positive test charge placed at that point.

- -- - -

- - - -- - -

- -

E

+ ++ + +

+ + + ++ + +

+ +

E


5

2o

e

q qF k

r

2e

qE k

r

Electric Field from a Point Charge

Suppose we have two charges, q and q0, separated by a distance r. The electric force between the two charges is

We can consider q0 to be a test charge, and determine the electric field from charge q as


6

+

r

qqo

E

-

r

qqoE

• If q is +ve, field at a given point is radially outward

from q.

• If q is -ve, field at a given point is radially inward from q.


7

Electric Field Lines

To visualize electric field patterns, one can draw lines pointing in the direction of the electric field vector at any point.These lines are called electric field lines.

1. The electric field vector is tangent to the electric field lines at each point.

2. The number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field in a given region.

3. No two field lines can cross each other . Why?


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The electric field lines for a point charge.

)a (For a positive point charge, the lines are directed radially outward .)b (For a negative point charge, the lines are directed

radially inward.

Note that the figures show only those field lines that lie in the plane of the page.


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The electric field lines for two positive point charges.

The electric field lines for two point charges of equal magnitude and opposite sign (an electric dipole)


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Question:

Two charges q1 and q2, fixed along the x-axis as shown, produce an electric field E at the point (x,y)=(0,d), which is the directed along the negative y-axis.

Which of the following is true?

1. Both charges are positive

2. Both charges are negative

3. The charges have opposite signs


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A system of two oppositely charged point particles is called an electric dipole.The vector sum of the electric field from the two charges gives the electric field of the dipole (superposition principle).We have shown the electric field lines from a dipole

Electric Field from an Electric DipoleElectric Field from an Electric Dipole


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ExampleExample::ExampleExample::

Two charges on the x-axis a distance

d apart Put -q at x = -d/2 Put +q at x = +d/2

Calculate the electric field at a point P a distance x from the origin


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Principle of superposition:The electric field at any point x is the sum of the electric fields from +q and -q

Replacing r+ and r- we get

2 20 0

1 1

4 4

q qE E E

r r

2

212

210

11

4 dxdx

qE

This equation gives the electric field everywhere

on the x-axis (except for x = d/2)


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Problem: (a) Find the electric field at point P in Fig.below. (b) Repeat for q2 = +1 nC.


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Example: Electric Field Due to Two Point Charges

Charge q1=7.00 C is at the origin, and charge q2=-10.00 C is on the x axis, 0.300 m from the origin. Find the electric field at point P, which has coordinates (0,0.400) m.

x

y

0.300 mq1 q2

0.40

0 m

P

E1

E2

E


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2

2

2

2

6

1 9 51 22

1

6

2 9 52 22

2

5325

541 2 1 25

2 2 5

1

7.00 108.99 10 3.93 10 /

0.400

10.00 108.99 10 3.60 10 /

0.500

2.16 10 /

sin 1.05 10 /

2.4 10 /

tan ( / ) 25.9

Nme C

Nme C

x

y

x y

oy x

CqE k N C

r m

CqE k N C

r m

E E N C

E E E E E N C

E E E N C

E E


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Problem : Electric Field Due to Four Point Charges

Four charges q1=10 nC, q2=-20 nC, q3=20 nC and q4=10 nC form a square of edge length 5 cm. What electric field

do the particles produce at the square center?


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ExampleIn Figure, determine the point (other than infinity) at which thetotal electric field is zero.

Solution: The sum of two vectors can be zero only if the two vectors have the same magnitude and opposite directions.


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Problem Three point charges are

arranged as shown in Figure

(a) Find the vector electric field that the 6.00 nc and –3.00 nc charges together create at the origin.

(b) (b) Find the vector force on the 5.00 nc charge.


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Motion of charge particles in a uniformelectric field

An electron moving horizontally passes between two horizontal planes, the upper plane charged negatively, and the lower positively. A uniform, upward-directed electric field exists in this region. This field exerts a force on the electron. Describe the motion of the electron in this region.

-ve

- - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + +

0E


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Horizontally: No electric field No force No acceleration Constant horizontal

velocity

0

0

0

x

x

x

x o

o

E

F

a

v v

x v t

Vertically: Constant electric field Constant force Constant acceleration Vertical velocity increase

linearly with time.2

/

/

1/

2

y o

y o

y o e

y o e

o e

E E

F eE

a eE m

v eE t m

y eE t m


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-

- - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + +

Conclusions: The charge will follow a parabolic path downward. Motion similar to motion under gravitational field only

except the downward acceleration is now larger.


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-e-e

-Q +Q

+Q

-Qx

Phosphor Screen

This device is known as a cathode ray tube (CRT)


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Continuous Charge Distributions

Ni

0 i02i 1 i0

qÊ k r

r

0 2all charge

dqÊ k r

r

Discrete chargesContinuous charge distribution

0 2

kqÊ r

r

Single charge

0 2

kdqˆdE r

r

Single piece of a charge distribution

+Q3

+Q2

+Q1

01E

03E

02E

10r

20r

30r 0 0

++

++

0dE

dq


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Q dq

dx

Line charge dq dx dq Rd

Cartesian Polar

Surface chargeQ dq

A dA dq dxdy dq rdrd

Volume chargeQ dq

V dV dq dxdydz dq rdrd dz

2dq r sin drd d

2

kdqˆdE r

r


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A rod of length l has a uniform positive charge per unit length λ and a total charge Q. Calculate the electric field at a point P that is located along the long axis of the rod and a distance a from one end.

Example: Electric Field Due to a Charged Rod

22 x

dxk

x

dqkdE

dxdq

ee

al

ae

al

a

e

al

a

e xk

x

dxk

x

dxkE

122

)(

11

ala

Qk

alal

QkE ee


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Example – Infinitely Long Line of Charge

+

+

+

+

dE

+

+

+

dy

x

dq dy

y

2 2 2r x y

2

kdqˆdE r

r

xdE

ydE

+

y-components cancel by symmetry

x 2

kdqdE cos

r

2 2 2 2

k dy xdE

x y x y

3 2

2 2 2

dy 2 2kE k x k x

x xx y


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Example – Charged Ring

dE

z

dq ds ad

2 2 2r z a

zdE

dE

a

d

+

+

+

+ +

+

+

2

kdqˆdE r

r

perpendicular-components cancel by symmetry

z 2

kdqdE cos

r

2 2 2 2

k ad zdE

z a z a

2

3 3 32 2 2 2 2 202 2 2

k za k za kQzE d 2

z a z a z a


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When: z a

The charged ring must look like a point source.

3 3 3 2

2 2 22 22

2

kQz kQz kQz kQE

z zz a a

z 1z

0


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Example – Uniformly Charged Disk

dE

z

3

2 2 2

kQzE

z r

r

3

2 2 2

kzdqdE

z r

dq dA rdrd 2 rdr

3

2 2 2

kz 2 rdrdE

z r

R

2 2

2

R R z R

3 3 32 2 2 20 02 2 2z

kz 2 rdr 2rdr duE kz kz

z r z r u

2 2

2 2

2

2

z R1

z R 3 22

2 2 2 2 2z

z

u 1 1 zkz u du kz 2kz k 2 1

1 z R z z R2


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Two Important Limiting Cases

2 2o o

z 1E k 2 1 k 2 2

4 2z R

Large Charged Plate: R z

dE

z

rR

Very Far From the Charged Plate: z R

12 2

22 2 2

2

z z RE k 2 1 k 2 1 k 2 1 1

zz R Rz 1

z

2 2 2

2 2 2 2

1 R 1 R k R kQk 2 1 1 k 2

2 z 2 z z z


32

Lecture 4

Discussion


33

[1 ]In figure, two equal positive charges q=2x10-6C interact with a third charge Q=4x10-6C. Find the magnitude and direction of the resultant force on Q.

0

46.0

0sinsin

46.023.02)5

4(29.0cos2

29.0)5.0(

)102)(104(109

2

669

21

1 2

NF

FFF

NFF

N

r

qQKFF

T

y

x

eQqQq


34

[2 ]A charge Q is fixed at each of two opposite corners of a square as shown in figure. A charge q is placed at each of the other two corners. If the resultant electrical force on Q is Zero, how are Q and q related.

12 13

2 2

13 14

2 2

0 cos 0

1.

2 2

....................................(1)2 20 sin 0

1.

2 2

....................................(1)2 2

2 2

x

y

F F F

kQq kQQ

a a

Qq

F F F

kQQ kQq

a a

Qq

then

Q q


35

[3 ]Two fixed charges, 1µC and -3µC are separated by 10cm as shown in figure (a) where may a third charge be located so that no force acts on it? (b) is the equilibrium stable or

unstable for the third charge?

31 32

3 1 3 22 2

31 32

6 6

22

22 2 2

1 10 3 10

10

3 10 3 20 100

e e

F F

q q q qK K

r r

d d

d d d d d


36

2 2

2

2 20 100 0 10 50 0

1. 10, 50

10 100 4(1)( 50)4

2 2

10 10 310 300 2

2 2

5 5 3 13.66

5 5 3 13.66

)the equilibrium unstable

d d d d

a b c

b b acd

a

d

d

d

b s


37

[4 ]Find the electric field at point p in figure due to the charges shown.

Solution: 1 2

4

3

4

2 2

36 10 /

28 10 /

( ) ( )

46.1 /

141

x

y

p x y

E E E

N C

E E

N C

E E E

N C


38

[5 ]A charged cord ball of mass 1g is suspended on a light string in the presence of a uniform electric field as in figure. When E=(3i+5j) *105N/C, the ball is in equilibrium at Θ=37o. Find (a) the charge on the ball and (b) the tension in the string.

Substitute T from equation (1) into equation (2)

5

5

3 10 /

5 10 /

sin 37 0....(1)

cos37 0....(2)

x

y

x x

y y

E N C

E N C

F qE T

F qE T

Substitute T from equation (1) into equation (2)

Substitute by q into equation (1) to find T=5.44*10-3N


39

[6 ]A 1.3µC charge is located on the x-axis at x=-0.5m, 3.2µC charge is located on the x-axis at x=1.5m, and 2.5µC charge is located at the origin. Find the net force on the 2.5µC charge

NFFF

r

qqKF

r

qqKF

t

e

e

3332321

32

669

223

3223

32

669

221

2121

10149103210117

10325.1

105.2102.3109

101775.0

105.2103.1109


40

[7 ]Two free point charges +q and +4q are a distance 1cm apart. A third charge is so placed that the entire system is in equilibrium. Find the location, magnitude and sign of the third charge. Is the equilibrium

stable?

md

md

d

dd

ddd

dd

d

qK

d

qK

EE

ee

3/1

16

13442

0123

012_4

14

1

4

2

22

22

22

21


41

[8] Two protons in a molecule are separated by a distance of 3.8*10-10m. Find the electrostatic force exerted by one proton on the other.

[9] The electric force on a point charge of 4.0mC at some point is 6.9*10-

4N in the positive x direction. What is the value of the electric field at that point?

9

21

9

210

999

221 106.1

104.14

106.14.14

108.3

106.1106.1109

r

qqKF e

CNq

FE 2

6

4

10725.1100.4

109.6


42

[10 ]Two point charges are a distance d apart . Find E points to the left P. Assume q1=+1.0*10-6C, q2=+3.0*10-6C, and d=10cm

2

6

2

69

21

2

69

22

22

2

69

21

11

10

100.3100.1109

10

100.3109

100.1109

xxEEE

xr

qKE

xr

qKE

PPT

eP

eP


43

]11[ Calculate E (direction and magnitude) at point P in Figure.

22

31

2

0

r

qkEE

EEE

ey

x

it fromaway triangle theof middle From

4.

4

1

2

2.

4

1

2

2

20

20

20

2

a

qE

a

q

a

q

a

qkEE eyT


44

[12 ]A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, 2.0cm away, in a time 1.5*10-8s. (a) What is the speed of the electron as it strikes the

second plate? (b) What is the magnitude of the electric field.

20

0

6

3

:

1/ 2

2.7 10 /

/ / 1 10 /

Horizental

first find the acceleration from the relation

x v t at

then find the velocity

v v at

v m s

then find the electric field E F q ma q N C


45

[13 ]Three charges are placed on corners of an equilateral triangle as shown in Figure 1. An electron is placed at the center of the

triangle. What is the magnitude of the net force on the electron?

NFFFF

iselectrontheonforcenettheand

Nr

cekF

isforcesthreetheofeachofmagnitudetheThus

mrr

yxnet922

92

106.82

103.41

,

577.05.0

30cos


46

[14 ]A uniform electric field exists in the region between two oppositely charged plane parallel plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate 2x10-8 s later. If the magnitude of the electric field

is 4x103 N/C, what is the separation between the plates ?

matd

bygivenisdistncetherestfromstartselectrontheSince

smm

eE

m

Fa

ee

14.0)102)(107(2

1

2

1

,

/107

28142

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General Physics II, Lec 3, By/ T.A. Eleyan 1 Lecture 3 The Electric Field