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Which graphical representation of electric field of a point charge
is correct?
A
B
_
C
D
_
E) I do not know
+
+

E
1
Q
rˆ
2
40 r
A

E
_
1
Q
rˆ
2
40 r
What if the magnitude of electric field exactly at the position
of a point charge?
r  0, E  
Point charge does not exert field on itself!
The Superposition Principle
The net electric field at a location in space is a vector sum
of the individual electric fields contributed by all charged
particles located elsewhere.
The electric field contributed by a charged particle is
unaffected by the presence of other charged particles.
The Superposition Principle

If many more point charges are present (qi) we go for i E i
and eventually for integral  !
System consists of two point charges (+4) and (+1)
+4
+1
How many points exist in space where E-field is equal zero?
A)
B)
C)
D)
E)
None
One
Two
Infinite number
I don’t know
E-field in the middle of uniformly positively charge ring is
represented by
y
What is about any other point
inside the ring?
x
A)
B)
C)
D)
Red arrow
Blue arrow
Zero
I do not know
The E of a Uniformly Charged Sphere
Can calculate using principle of superposition:

1 Q
Esphere 
rˆ
2
40 r

Esphere  0
for r>R (outside)
for r<R (inside)
The Superposition Principle
The electric field of a dipole:
Electric dipole:
Two equally but oppositely charged
point-like objects
s
-q
+q
Example of electric dipole: HCl molecule
What is the E field far from the dipole (r>s)?
Dipoles are ubiquitous
1. Atom will become dipole when it experiences electric field:
2. Some molecules exist in the form of permanent dipoles:
HCl molecule
In Protein the carbonyl groups of
polypeptides have substantial
dipole moments and their
electrostatic interactions make an
important energetic contribution to
the stability of an a-helix
O C
+
D=3.7 Debye
Calculating Electric Field
Choice of the origin
y
s
-q
+q
x
z
Choice of origin: use symmetry
1. E along the x-axis
E1, x  E , x  E , x 
E1, x 
E1, x 
1
q
4 0 r  s 2 
2

1
q
4 0 r  s 2 
2
1 qr 2  qrs  qs 2 / 4  qr 2  qrs  qs 2 / 4
4 0
1
r  s 2  r  s 2
2
2qrs
4 0 r  s 2 2 r  s 2 2
2
Approximation: Far from the Dipole
E1, x 
1
2 srq
2
2
40 
s 
s
r

r


 

2 
2

2
2
s 
s

if r>>s, then  r     r    r 2
2 
2

E1, x
1 2 sq

40 r 3

E1 
1 2sq
,0,0
3
40 r
While the electric field of a point charge is proportional to 1/r2,
the electric field created by several charges may have a different
distance dependence.
2. E along the y-axis

E
1
q
rˆ
2
40 r


s
s
r  0, y ,0  ,0,0   , y ,0 
2
2

  r  r 

s
s
r  0, y ,0   ,0,0  , y ,0 

2
2
s
y2   
2
2

E 

E 
1
q
40
s
y  
2
s
 , y ,0
2
2
2
1
q
40
s
y2   
2
s
y2   
2
2
s
, y ,0
2
2
s
y2   
2
2
s
y2   
2
2
2. E along the y-axis

E 
1
q
40
s
y  
2
2
s
 , y ,0
2
2
s
y2   
2
2



1
E2  E   E  
40
s
y2   
2
2

E 
1
q
40
s
y  
2
2
qs
q
33
2 2
 2 s
y   
2

s
 , y ,0
2
2
s
y  
2
2
2
 1s,0,0



1 qs
,0,0
if r>>s, then E2 
3
4 0 r
at <0,r,0>
3. E along the z-axis

E1 

E2 
 1 qs
,0,0
3
40 r
1 2sq
,0,0
3
40 r
at <0,r,0>
or <0,0,r>
Due to the symmetry E along the z-axis must
be the same as E along the y-axis!
at <r,0,0>
Other Locations
The Electric Field
Point Charge:

E1 
1
q1
rˆ
2
40 r
Dipole:
for r>>s :
y
+q
z
1 2qs
E
,0,0
3
4 0 r
x
-
at <r,0,0>
1 qs
,0,0
3
4 0 r
at <0,r,0>
1 qs
E
,0,0
3
4 0 r
at <0,0,r>
E
s
-q
+
System consists of two point charges (+1) and (-1)
+1
-1
How many points exist in space where E-field is equal zero?
A)
B)
C)
D)
E)
None
One
Two
Infinite number
I don’t know
Dipole in a Uniform Field


F  qE
Forces on +q and –q have the same
magnitude but opposite direction



Fnet  qE  qE  0
It would experience a torque about its
center of mass.
What is the equilibrium position?
Electric dipole can be used to measure
the direction of electric field.
Dipole Moment
x:
y, z:

E1 

E2 
1 2qs
p
,0
,0,0
,0
33
40 rr
r>>s
 1 qs
p
,,00,,00
33
40 r
The electric field of a dipole is proportional to the
Dipole moment: p = qs

p  qs, direction from –q to +q
Dipole moment is a vector pointing
from negative to positive charge
Electric Field
 
E  F /q
 
E  E  x, y , z , t 
Electric field has units of Newton per Coulomb:
[N/C]
A Fundamental Rationale
• Convenience: know E at some
r location – know the electric
force on any charge: F  qE
• Can describe the electric properties of matter in terms of
electric field – independent of how this field was produced.
Example: if E>3106 N/C air becomes conductor
• Retardation
Nothing can move faster than light c
c = 300,000 km/s = 30 cm/ns
Coulomb’s law is not completely correct – it does not
contain time t nor speed of light c.

F
1 q1q2
rˆ
2
40 r

E
1
q
rˆ
2
40 r
v<<c !!!
Example Problem
y
E=?
A dipole is located at the origin, and is composed of particles with
charges e and –e, separated by a distance 210-10 m along the xaxis. Calculate the magnitude of the E field at <0,210-8,0> m.
sq
Since r>>s:
40 r 3
2 
10
19 


Nm
2

10
m

1.6

10
C
9
E1, x   9  10


3
2 
8
C  


2  10 m
N
E1, x  7.2  104
C
E1, x 
200Å
1

x
2Å
Using exact solution:
4 N
E1, x  7.1999973  10
C

Interaction of a Point Charge and a Dipole

Edipole

F

F


F  QEdipole  Q
 1 2qs
,0,0
3
40 d
• Direction makes sense?
- negative end of dipole is closer, so its net contribution is larger
• What is the force exerted on the dipole by the point charge?
- Newton’s third law: equal but opposite sign