General Physics I
Download
Report
Transcript General Physics I
Electric dipole,
systems of charges
Physics 122
7/22/2015
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
7/22/2015
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
7/22/2015
Lecture III
3
Concepts
• Primary concepts:
– Electric field
• Secondary concepts:
– Electric dipole
7/22/2015
Lecture III
4
Laws
• Dipole field
• Dipole in electric field: energy and
torque
• Superposition principle for a continuous
distribution of charge
7/22/2015
Lecture III
5
Skills
• Calculate electric field of a system of charges
7/22/2015
Lecture III
6
Electric field
F k
Q 1Q 2
r
2
F Q1 E
F Q1 E
+
1
E
7/22/2015
• F – force between two charges(N)
• Q – electric charge (C= Coulomb)
E k
Q2
r
+
2
2
• 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
Lecture III
7
Superposition of fields
E E1 E 2
E1
+
E2
Positive test charge
+
-
1
2
7/22/2015
Principle of superposition:
Net field created by a
system of charges is a
vector sum of fields
created by individual
charges:
E E 1 E 2 E 3 ....
Lecture III
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
7/22/2015
Lecture III
9
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
p
x
3/2
L 2
2
2
... for r >> L,
E k
7/22/2015
Lecture III
p
r
3
10
Torque
Force makes objects move torque makes objects rotate
rF rF sin
7/22/2015
Lecture III
11
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
7/22/2015
Lecture III
12
How to add torques?
Axis of rotation
1 + 2
F2
F1
Axis of rotation
1 - 2
F2
F1
7/22/2015
Lecture III
13
How to add torques?
Axis of rotation
F2
1 - 2
F1
Axis of rotation
1 + 2
F2
F1
7/22/2015
Lecture III
14
Electric dipole
• Dipole in uniform E
• Net force F=F+-F-=0
• Net torque
F
l
2
sin F
l
sin Fl sin
2
F QE
Fl sin EQl sin Ep sin
E p
7/22/2015
Lecture III
15
Electric dipole
• Dipole in uniform E
• Energy - ?
• Work done by the field
2
W
d
1
2
Ep
sin d
pE (cos 2 cos 1 )
1
U W pE cos p E
7/22/2015
Lecture III
16
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
7/22/2015
Lecture III
17
Dipole in electric field
E external
E int ernal
+Q
-Q
E net E external E int ernal E external
• In electric field dipoles line up with the field
• Dipole internal field anti-parallel in external field
• Net field is reduced
7/22/2015
Lecture III
18
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).
7/22/2015
Lecture III
19
The Electric Field of
a system of charges
Bunch of Charges
Charge Distribution
E E1 E2 E3 ...
Ei k
qi
ri
2
dE k
, ri - distance from charge i to
point in space where E is evaluated
+
+
-
+
+
-
-
dq
r
E dE
2
+ +++ + +
+ + +++
+
+
7/22/2015
Lecture III
20
Vectors by components
Charge Distribution
dE k
dq
r
E dE
+ +++ + +
+ + +++
dE
2
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
7/22/2015
Lecture III
21
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
7/22/2015
Lecture III
22
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
Ek
7/22/2015
2
a
2
Ex
x
Qx
x a
E
E
(x a )
2
2 3/ 2
Lecture III
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 :
E 2pks
7/22/2015
s
2 0
Lecture III
24
Two parallel plates
• Infinite plates
• One positive, one
negative,
• Same charge density s
s
E
2 0
s
E
2 0
s
E E E
0
+
7/22/2015
Lecture III
25
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).
dE k
dq
r
dE x k
k
2
x y
r
x y
2
2
x y
2
2
y
ldy
ldy
2
dq=ldy
dE x
x
2
dE y
cos
E 2k
l
x
7/22/2015
Lecture III
26