1/16/07184 lecture 51 phy 184 spring 2007 lecture 5 title: electric field examples

1/16/07 184 Lecture 5 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 5

Title: Electric Field Examples

1/16/07 184 Lecture 5 2

AnnouncementsAnnouncementsAnnouncementsAnnouncements

Homework Set 1 is done. The average score was 9.6/10.

Homework Set 2 is open - we will open Homework Sets now already on Thursday.

Helproom coverage is posted in LON-CAPA.

Honors option work in the SLC:

• Please sign up for time slots! Use the sign-up sheet after class.

1/16/07 184 Lecture 5 3

Review - The Electric FieldReview - The Electric FieldReview - The Electric FieldReview - The Electric Field

A charge creates an electric field around itself and the other charge feels that field.

Test charge: point object with a very small positive charge so that it does not modify the original field

+

The electric field at a specified point: place a positive test charge q at the point and measure the electrostatic force that acts on the test charge,

Test charge q +

qF

xE

)(

1/16/07 184 Lecture 5 4

Review -Field Lines from a Point Review -Field Lines from a Point ChargeCharge

Review -Field Lines from a Point Review -Field Lines from a Point ChargeCharge

The electric field lines from a point charge extend out radially.

For a positive point charge, the field lines point outward• Terminate at infinity

For a negative charge, the field lines point inward• Originate at infinity

1/16/07 184 Lecture 5 5

Electric Field from a Point ChargeElectric Field from a Point ChargeElectric Field from a Point ChargeElectric 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

The electric field is a vector, so to add electric fields we must add the components separately.

20

04

1

r

qqF

200 4

1

r

qqF

E

1/16/07 184 Lecture 5 6

Electric Field from 4 Point Charges (1)Electric Field from 4 Point Charges (1)Electric Field from 4 Point Charges (1)Electric Field from 4 Point Charges (1) 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?

Idea: Use the superposition principle.

Step 1: Choose your coordinate system and stick with it!

Step 2: Look at the x and y coordinates separately.

4321 EEEEE

1/16/07 184 Lecture 5 7

Electric Field from 4 Point Charges (2)Electric Field from 4 Point Charges (2)Electric Field from 4 Point Charges (2)Electric Field from 4 Point Charges (2)

23

3

22

2

41 0 and 0

r

qkE

r

qkE

EE

x

x

xx

x component (at 0)

E2x is positive!

E3x is positive!

q1 = 10 nC, q2 = -20 nCq3 = 20 nC, q4 = -10 nC

C

N1088.2

C

N

2/)05.0(

1040109 5

2

99

xE

1/16/07 184 Lecture 5 8


24

4

21

1

32 0 and 0

r

qkE

r

qkE

EE

y

y

yy

y component (at 0)

E1y is negative!

E4y is negative!

q1 = 10 nC, q2 = -20 nCq3 = 20 nC, q4 = -10 nC

C

N1044.1

C

N

2/)05.0(

1020)(109 5

2

99

yE

1/16/07 184 Lecture 5 9


E at the center pt.

axisx below the degrees

)/Earctan(E directionC

N105.3 magnitude

xy

522

yx EEE

1/16/07 184 Lecture 5 10

Electric Field from Three Point Electric Field from Three Point ChargesCharges

Electric Field from Three Point Electric Field from Three Point ChargesCharges

Consider three charges

The three charges are placed at

What is the electric field at point P ?

P: (b,a)

C 5.3 C 5.2 C 5.1 321 qqq

)0,(: )0,0(:

),0(:

32

1

bqq

aq

a = 8.0 m ; b = 6.0 m

1/16/07 184 Lecture 5 11

Electric Field from Three Point Charges (2)Electric Field from Three Point Charges (2)Electric Field from Three Point Charges (2)Electric Field from Three Point Charges (2)

The electric field at P due to q1 is



xb

qkE ˆ

21

1 x

b

qkE ˆ

21

1

ya

qkE ˆ

23

3

yba

kqx

ba

kqE ˆsinˆcos

222

222

2

Note: tan = a/b

1/16/07 184 Lecture 5 12

Now we add the components, to obtain

Magnitude of E

Direction of E

Electric Field from Three Point Charges (3)Electric Field from Three Point Charges (3)Electric Field from Three Point Charges (3)Electric Field from Three Point Charges (3)

N/C 311

N/C 509

Ey

Ex

N/C 59722 yx EEE

N/C 311

N/C 509

y

x

E

E

5.31)/arctan( axis with x angle xy EE

1/16/07 184 Lecture 5 13

Electric Field from an Electric DipoleElectric Field from an Electric DipoleElectric Field from an Electric DipoleElectric Field from an Electric Dipole

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

1/16/07 184 Lecture 5 14

Electric Field from an Electric Dipole Electric Field from an Electric Dipole (2)(2)


Expression for the electric field of a dipole along a line including both charges …

We will derive a general expression good anywhere along the dashed line and then get an expression for the electric field a long distance from the dipole.

1/16/07 184 Lecture 5 15



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

1/16/07 184 Lecture 5 16



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

20

20 4

1

4

1

r

q

r

qEEE

2

212

210

11

4 dxdx

qE

1/16/07 184 Lecture 5 17



This equation gives the electric field everywhere on the x-axis (except for x = d/2)

Let’s look at this equation far away along the positive x-axis (x >> d)

2

212

210

11

4 dxdx

qE

30

20

20

2

2

4

114

x

qdxd

x

q

xd

xd

x

qE

Taylor series

21

1

12

1/16/07 184 Lecture 5 18

Definition of Electric Dipole MomentDefinition of Electric Dipole MomentDefinition of Electric Dipole MomentDefinition of Electric Dipole Moment

We define the vector electric dipole moment as a vector that points from the negative charge to the positive charge• p is the magnitude of the dipole moment• q is the magnitude of one of the opposite

charges• d is the distance between the charges

Using this definition we can write the electric field far away from an electric dipole as

dqp

302 x

pE

1/16/07 184 Lecture 5 19

Functional Dependence E(x) Functional Dependence E(x) Functional Dependence E(x) Functional Dependence E(x)

1/4

1/8

E(x)

E(x)=E(x)=

Point charge

Dipole

1/16/07 184 Lecture 5 20

Electric Dipole Moment of a Water MoleculeElectric Dipole Moment of a Water MoleculeElectric Dipole Moment of a Water MoleculeElectric Dipole Moment of a Water Molecule

Chemistry reminder - the H2O molecule

The distribution of electric charge in a H2O molecule is non-uniform. The more electronegative oxygen atom attracts electrons from the hydrogen atoms. Thus, the oxygen atom acquires a partial negative charge and the hydrogen atoms acquire a partial positive charge. The water molecule is "polarized."

1/16/07 184 Lecture 5 21

Example - Electric Dipole Moment of WaterExample - Electric Dipole Moment of WaterExample - Electric Dipole Moment of WaterExample - Electric Dipole Moment of Water

Suppose we approximate the water molecules as two positive charges located at the center of the hydrogen atoms and two negative charges located at the center of the oxygen atom. What is the electric dipole moment of a water molecule?

1/16/07 184 Lecture 5 22

Electric Dipole Moment of Water (2)Electric Dipole Moment of Water (2)Electric Dipole Moment of Water (2)Electric Dipole Moment of Water (2)

Our result for the electric dipole moment of water is then

This oversimplified result is comparable to the measured value of 6.2x10-30 C m.(The assumed charge distribution not precise.)

mC 1022

m 106.05.52cosm 1029

1010

edp

d

1/16/07 184 Lecture 5 23

Demo - PolarizationDemo - PolarizationDemo - PolarizationDemo - Polarization

1/16/07 184 Lecture 5 24

Math Reminder (1)Math Reminder (1)Math Reminder (1)Math Reminder (1)

a

a

d

a

b

cx

y

1/16/07 184 Lecture 5 25

Math Reminder (2)Math Reminder (2)Math Reminder (2)Math Reminder (2)

Binomial theorem :

(Taylor series)

In our case: x=d/2x, n=-2 and x=-d/2x, n=-2

1/16/07184 lecture 51 phy 184 spring 2007 lecture 5 title: electric field examples

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