Lecture 2-1

COULOMB FORCE

1 22

ˆq q

F k rr

Charge q1 “Test Charge” q2

r1,2

1,21,2

Question: How can q1 sense q2 without direct contact?

• NO! just as air around us, there’re some medium around a charge, the electric field that transfers the force.

“Is action at a distance possible?”

Lecture 2-2

READING QUIZ 1

HOW DOES THE MAGNITUDE OF THE ELECTRIC FIELD E FROM A POINT CHARGE Q DEPEND ON THE MAGNITUDE OF Q AND THE DISTANCE R TO THE FIELD POINT?

A) E = k Q Q / R2

A) E = k Q / R

B) E = k Q / R2

C) E = k Q Q / R

A) E = k Q / R3

Lecture 2-3

Electric Field

Define electric field, which is independent of the test charge, q2 , and depends only on position in space:

FE

q��������������

��������������

20

1ˆ

4

F QE r

q r

Electric Field due to a Point Charge Q

Lecture 2-4

Electric Field Lines

• A visualization tool to illustrate the geometry of an electric field.

• Electric field lines originate from positive charges and terminates at negative charges.

• The direction of the electric field at any location is tangential to the field line there.

• The magnitude of the electric field at any location is proportional to the density of the lines there.

# lines/area ~ 1/r2 ~ |E|

(or at infinity)

Lecture 2-5

Electric Field

With this concept, we can “map” the electric field anywhere in space

A Vector field

77

82

8368

5566

83

75 80

9091

7571

80

72

84

73

57

8892

77

5688

7364

Lecture 2-6

Demonstration

• Vandegraaff Generator

Lecture 2-7

Electric field lines of two charges of equal magnitude

dipole

Far from the charges, electric field lines resemble those of a point charge.

Field lines do NOT cross

They do NOT come out of nowhere or vanish into a point.

Lecture 2-8

Two charges of the same magnitude

• Both charges > 0

• One is > 0, the other < 0

-q q

electric dipole of dipole moment:

p q d����������������������������

d��������������

if |qR|>|qL| if |qL|>|qR|

3

2kpE

x

Lecture 2-9 Electric field lines of two unequal charges

# lines proportional to the magnitude of charge

Far from charges, the field lines are as if they are due to a point charge of +2q-q=+q

http://www.its.caltech.edu/~phys1/java/phys1/EField/EField.html

http://www.falstad.com/vector3de/

Lecture 2-10

©2008 by W.H. Freeman and Company

Lecture 2-11

Electric Field due to Multiple Point Charges

The force on a test charge is then given by

321 FFFF

so the electric field is, by definition, given by

321

0

3

0

2

0

1

0

EEE

qF

qF

qF

qF

E

Principle of Superposition!

Lecture 2-12

Lecture 2-13

Lecture 2-14

Σ Fx = 3.97 x 10-7 N Σ Fy = - 2.77 x 10-7 N

F2 = Fx2 + Fy2 = (4.84 x 10-7 N)2

Tan Θ = Fy /Fx = - 34.9 O

( Sum symbol Σ ) E = F/qO

Lecture 2-15Warm-up quiz 2

What is the direction of the electric field at the center of the two point charges? (Take Q > 0.)

a) toward the lower rightb) toward the upper leftc) upwardd) to the righte) none of the above

Q -Q

Lecture 2-16

Point Charge in an external Electric Field

• Electrostatic force on the charge q due to E:

• E acting on the charge is produced by the other charges (external field).

• Dimension of E = [force]/[charge] and

its SI units = N/C

• Work done by E on q in moving the charge:

F qE����������������������������

F is parallel to E if the charge is positive and anti-parallel if the charge is negative

The charge is not affected by its own electric field.

:A B A BW qE dl

���������������������������� W is path( )-independent

=> Conservative force

Lecture 2-17

Demonstration

• Dynamics of a charged Mass in an Electric field

Lecture 2-18

Dynamics of a Charged Mass in Electric Field

For -Q<0 in uniform E downward:

2

2 2

2

( )

( )

1( ) , ( )

2

1

2 2

y

y x

yx x

F ma Q E

QEa a j j E E j

m

y t a t x t v t

x QExy a

v mv

������������������������������������������

������������� �

22 2 2( ) ( )x y x

QEtv t v v t v

m

• Oscilloscope

• Ink-Jet Printing

• Oil drop experiment

tan y

x

v

v

-Q

http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/electric_single_particle/applet.html

vy = at = qE/m t vx >>0

Lecture 2-19

The Electric Dipole

Along the + z-axis

Where p=qd is the magnitude of the dipole moment. The electric dipole is a vector in the direction

from –q to +q. Let d 0 keep qd constant.This a point dipole

Lecture 2-20

Dipole in a uniform external electric field

• No net force. The electrostatic forces on the constituent point charges are of the same magnitude but along opposite directions. So, there is no net force on the dipole and thus its center of mass should not accelerate.

• Net torque! There is clearly a net torque acting on the dipole with respect to its center of mass, since the forces are not aligned.

Clockwise about the center of mass

http://qbx6.ltu.edu/s_schneider/physlets/main/dipole_torque.shtml

Lecture 2-21

Torque on the Dipole

The magnitude of the torque:

sin

sinsin

sin)(sin

pE

qEdFd

xdFFx

The torque points into the screen.

Ep

x

F

F

CM

Lecture 2-22

Electric Dipoles

• Typical dipole consists of positive and negative charges slightly displaced.

• General definition of dipole moments exists: )3( )p r rd r

����������������������������

• Water molecule can be thought of as consisting of 2 standard dipoles at an angle to each other.

Net neutral molecules can have electrical dipole moments

Permanent dipole moment (polar) vs. induced dipole moment

Lecture 2-23Physics 241 – 10:30 Quiz 3 – January 14, 2010

Four point charges are arranged at the corners of a square as shown. What is the direction of the electric field at the center of the square? (Take Q > 0.)

a) toward the lower rightb) toward the upper leftc) upwardd) to the righte) none of the above

- Q -Q

Q Q

Lecture 2-24Physics 241 – 11:30 Quiz 3 – January 14, 2010

Four point charges are arranged at the corners of a square as shown. What is the direction of the electric field at the center of the square? (Take Q > 0.)

a) toward the lower rightb) toward the upper leftc) upwardd) to the righte) none of the above

- Q Q

Q Q

Lecture 2-25Physics 241 – Sample Quiz C – January 9, 2008

Three point charges are arranged at three of the corners of a square as shown. What is the direction of the electric field at the center of the square? (Take Q > 0.)

a) toward the lower rightb) toward the upper leftc) upwardd) to the righte) none of the above

Q Q

Q