Lecture 2-1

Electric Field

1 22

ˆq q

F k rr

q1 q2

r1,2

1,21,2

Question: Why q1 can feel q2 without direct contact?

• just as air around us, there’re something around a charge, i.e. electric field (photons) that transfer the force.

Lecture 2-2

Electric Field

Define electric field, which is independent of the test charge, q, 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

Direction of Electric Field•point away from positive charges•Point to negative charges.

Q q F

Lecture 2-3

Electric Field

With this concept, we can “map” the electric field anywhere in space produced by any arbitrary charge: a Vector field

77

82

8368

5566

83

75 80

9091

7571

80

72

84

73

57

8892

77

5688

7364

Lecture 2-4

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!

++

+

+q0

F1

F2

F3

Lecture 2-5

Example: 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

x

Lecture 2-6

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 2-7

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 Electric field lines of two 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 2-9Warm-up quiz

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 2-10

Point Charge in an 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.

W is path( )-independent

=> Conservative force

dlEqWBABA :

Lecture 2-11

Dynamics of a Charge 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

Lecture 2-12

Electric Field due to an Electric Dipole

( ) ( )

2 2

2 22

2

2

2 3

( / 2) ( / 2)

(1 ) (1 )2 2

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

2 2[1 ( )] ...

E E E

q qk k

z d z d

q d dk

z z z

q d dk

z z z

kq d d kO

z zp

z z

At point P on z-axis:

Where is the (magnitude of) dipole moment.

p qd

Lecture 2-13

Dipole Field Anisotropy

( ) ( )

2 2

2 2 3/ 2

23/ 2

3 2

2

3 2

3

2 sin( / 2)

2[ ( / 2) ] 2

(1 )4

3(1 ...)

8

...

zE E E

qk

y d

q dk

y d

qd dk

y y

qdk

p

d

y y

k

y

At point P on y-axis: y

z

Dipole Field at General Point P:

where is the unit vector from the center of dipole to the observation point P.

n

3

ˆ)ˆ(3

r

pnnpE

Lecture 2-14

Electric Dipoles

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

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

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• 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 2-15

Dipole in uniform electric fields

• 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 2-16 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

Fr

Lecture 2-17Physics 241 – Sample Quiz A – January 9, 2008

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 2-18Physics 241 – Sample Quiz B – January 9, 2008

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 2-19Physics 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