Download General Physics I

Electric dipole,
systems of charges
Physics 122
5/25/2017
Lecture III
1
Workshops
• Due to low interest – 4 people and very limited
resources I have to cancel one of the workshops:
• Fridays, 4-6 pm B&L 108A
• Please let me know alternative times I’ll switch
you to other workshops
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Lecture III
2
I am running
Rochester marathon
•
•
•
•
This Saturday, September 17, 8:00 am
http://www.rochestermarathon.com/race.htm
Starts and ends at Frontier field
Goes along East and returns on Park Ave
– Lots of coffee shops and sit back, relax and watch
people suffer
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Lecture III
3
Concepts
• Primary concepts:
– Electric field
• Secondary concepts:
– Electric dipole
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Laws
• Dipole field
• Dipole in electric field: energy and
torque
• Superposition principle for a continuous
distribution of charge
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5
Skills
• Calculate electric field of a system of charges
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6
Electric field
Q1Q2
F k 2
r


F  Q1 E

E
+
+
1
2
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• F – force between two charges(N)
• Q – electric charge (C= Coulomb)


F  Q1  E
Q2
Ek 2
r
• E – electric field created at point 1
by charge 2
• Charge 2 has changed the property
of space at point 1
• Charge 1 is experiencing this
change
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Superposition of fields
  
E  E1  E2

E1
+
Positive test charge

E2
+
-
1
2
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Principle of superposition:
Net field created by a
system of charges is a
vector sum of fields
created by individual
charges:
   
E  E1  E2  E3  ....
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8
Electric dipole
• Two opposite charges of
equal value Q separated by
distance l
• Define dipole moment:


p
-Q
l

p  Ql
+Q
p1
• A vector directed from
negative charge to positive.
• Example – water molecule
H+
-O
-
p
H+
p2
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Electric Dipole Field Lines
y
• Lines leave positive charge
and return to negative charge
What can we observe about E?
x
• Ex(x,0) = 0
• Ex(0,y) = 0
• Field largest in space between two charges
• We derived:
E y x,0  k
... for r >> L,
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Lecture III
p
x    
2
p
Ek 3
r
3/ 2
L 2
2
10
Torque
Force makes objects move  torque makes objects rotate
  rF  rF sin 
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How to add torques?
• You have to think…
– If the force acts to rotate the system
• counterclockwise – torque and angular
acceleration are positive
+
• clockwise – torque and angular
acceleration are negative
-
• Only relative sign matters
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How to add torques?
Axis of rotation
F2
F1
Axis of rotation
  1 +  2
  1 -  2
F2
F1
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How to add torques?
Axis of rotation
F2
F1
  1 -  2
Axis of rotation
  1 +  2
F2
F1
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Electric dipole
• Dipole in uniform E
• Net force F=F+-F-=0
• Net torque
l
l
  F sin   F sin   Fl sin 
2
2
F  QE
  Fl sin   EQl sin   Ep sin 
 
  E p

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Electric dipole
• Dipole in uniform E
• Energy - ?
• Work done by the field
2
2
1
1
W   d   Ep  sin d  pE (cos  2  cos 1 )
 
U  W   pE cos    p  E
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Energy of dipole in electric field
U   pE cos 

p
-Q

p
+Q
+Q

p
+Q
-Q
-Q
 0
  90
U   pE
U 0
o
  180
o
U  pE
• Lowest energy state – dipole parallel to the field
• In electric field dipoles line up with the field
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Dipole in electric field
Eexternal

Eint ernal
+Q



 Eexternal  Einternal  Eexternal
-Q

Enet
• In electric field dipoles line up with the field
• Dipole internal field anti-parallel in external field
• Net field is reduced
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Test question
If this region is filled with pure water
(an excellent insulator), does the
electric field…
A) Increase?
B) Decrease?
C) Remain the same
+
+
+
+
+
E
-
+
+
+
+
+
E
-
The positive charge is shielded by the negative charges of
the aligned dipoles (and vice versa).
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The Electric Field of
a system of charges
Bunch of Charges
Charge Distribution
  

E  E1  E2  E3  ...
qi
Ei  k 2 , ri - distance from charge i to
ri

point in space where E is evaluated
+
+
-
+
+
-
-
dq
dE  k 2
r


E   dE
+ +++ + +
+ + +++
+
+
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Vectors by components
Charge Distribution

dE
dq
dE  k 2
r


E   dE
+ +++ + +
+ + +++

r
dE y  dE sin 
dEx  dE cos 
dq
r,  are different for different charges and
depend on your definition of the coordinate system,
So choose it wisely
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Symmetry and coordinate systems
• Coordinate systems are there to help you
• You have a choice of
– System type
• Cartesian
• Cylindrical
• Spherical
– Origin (0,0), Direction of axis
• A good choice (respecting the symmetry of the
system) can help to simplify the calculations
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Ring of charge
• A thing ring of radius a holds
a total charge Q. Determine
the electric field on its axis, a
distance x from its center.
r  x2  a2
a
Ex
x
Qx
E  k 2 2 3/ 2
(x  a )
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Lecture III

E

E
23
Charged disk
• Disk of radius R, uniformly charged
with Q, determine E on the axis, a
distance z above the center.
• Define charge density
s =Q/pr2
• Reuse previous results – divide disk
into rings radius r, integrate over r
from 0 to R.
z


z
E  2pks 1  2
2 1/ 2 
 (z  R ) 
z  R :
s
E  2pks 
2 0
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Two parallel plates
• Infinite plates
• One positive, one
negative,
• Same charge density s

s
E 
2 0
 

s
E  E   E 

s
E 
2 0
0
+
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Long line of charge
Determine the magnitude of the
electric field at a distance x from
a very long wire of uniformly
distributed charge with linear
charge density l (C/m).
dq=ldy
r  x2  y2
y
dq
ldy
dE  k 2  k 2 2
r
x y
ldy
dE x  k 2 2 cos 
x y
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dE x

x
dE y
l
E  2k
x
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