Download Document

Comments: Yours and Ours
Please go over the integrations one more time and please explain how much
mathematical knowledge is expected in this class.
Will we need to know how to write an integral like the one shown in the prelecture or
in the Question in the prelecture? If so, can we focus a bit more on what each piece
means physically?
The integrals, they went by too quickly
I found infinite lines of charge to be slightly confusing in terms of the integration
involved.
the balloon popping thing was awesome!!! also what the heck is a pre-flight? i know
what a pre-lecture is, we had plenty of those in 211, but pre-flight? is that the same as
Checkpoint or something?
The idea of "force per unit charge" is a little hard to get used to, especially in the idea
of which q is which when we write " F/q = K*(q/r^2). Also, what is the r with the pointy
accent thing over it at the end of the force equation? It isn't mentioned and doesn't
seem to do anything...
Electric fields are pretty powerful things. You should go in to more detail about how
awesome they are.
1.
2.
3.
Please respect your fellow students - please operate only your own clicker
Quite a bit of homework this week and building will be closed Sun/Mon 
Don’t panic!......Don’t be intimidated by integrals!
Physics 212 Lecture 2, Slide 1
Electricity & Magnetism
Lecture 2
Today’s Concepts:
A) The Electric Field
B) Continuous Charge Distributions
I find that relating everything to integrals made me very confused. The concept of electric fields
also baffles me - is it simply just a method of quantifying the force from a charge of a specified
amount onto another a specified distance away?
05
Electricity & Magnetism Lecture 2, Slide 2
Coulomb’s Law (from last time)
If there are more than two charges present, the total force on any
given charge is just the vector sum of the forces due to each of the
other charges:
q2
F4,1
F4,1
q1
F1
F2,1
q3
F3,1
q4
F2,1
F2,1
F3,1
 kq q
F1  r
F4,1
09
+q1 -> -q1  direction reversed
kq1q3
kq1q4
rˆ + 2 rˆ13+ 2 rˆ14
r13
r14
1 2
12
2
12
q4
F1
q3
F1
F2,1
q1
F1
F3,1
MATH:
q2
F3,1
 F kq kq
E  q  r rˆ + r rˆ
1
1
F4,1
2
2
12
12
3
2 13
13
+
kq4
rˆ
2 14
r14
Electricity & Magnetism Lecture 2, Slide 3
Electric Field
“Can you explain the derivations of the equations for electric fields? “
“What is the essence of an electric field? “

The electric field E at a point in space is simply
the force per unit charge at that point.
Electric field due to a point charged particle
Superposition

Qi
E   k 2 rˆi
ri
i
 F
E
q

Q
E  k 2 rˆ
r
q2
E4
E2
E
Field points toward negative and
Away from positive charges.
E3
q4
q3
12
Electricity & Magnetism Lecture 2, Slide 4
CheckPoint
“I can't figure out the field directions at all.”
y
A
x
+Q
-Q
B
Two equal, but opposite charges are placed on the x axis. The positive charge is placed to
the left of the origin and the negative charge is placed to the right, as shown in the figure
above.
What is direction at point A
a) Up b) down c) Left d) Right e) zero
15
What is direction at point B
a) Up b) down c) Left d) Right e) zero
Electricity & Magnetism Lecture 2, Slide 5
CheckPoint
E
+Q
A
+Q
A
E
-Q
Case A
“in case B, the two charges
carry same charges, they will
counteract in the line parallel to
the line connecting them”
+Q
In which of the two cases
shown below is the
magnitude of the electric
field at the point labeled A
the largest? (Select C if
you think they are equal)
“Same because
the forces are not
canceling each
other out”
Case B
“The electric field of
the point charge is
kq/r^2 so if two point
charges are the same
sign the field will be
greater”
A
18
B
Equal
Electricity & Magnetism Lecture 2, Slide 6
Two Charges
Two charges q1 and q2 are fixed at points (-a,0) and (a,0) as shown.
Together they produce an electric field at point (0,d) which is directed
along the negative y-axis.
y
(0,d)
E
(-a,0)
q1
q2
(a,0)
x
Which of the following statements is true:
A) Both charges are negative
B) Both charges are positive
C) The charges are opposite
D) There is not enough information to tell how the charges are
related
20
Electricity & Magnetism Lecture 2, Slide 7
+
+
+
21
_
_
_
Electricity & Magnetism Lecture 2, Slide 8
CheckPoint
A
B
C
INTERESTING: statement is correct, but given
in support of “to the left” !!
D
“(A LEFT)In Coulomb's law, the distance is squared, so
a doubling of distance is more significant than a
doubling of charge. “
“(B RIGHT)The distance is squared, so the charge
on the right hand side would need to be 4 times
as large for the particle to remain still..”
“(C Still) The +2Q is 2r away from the q so the 2 will
cancel out and just be +Q and r which is the same
as on the left.”
24
Electricity & Magnetism Lecture 2, Slide 9
Example
“Show me more electric field examples, please!”
What is the direction of the electric field
at point P, the unoccupied corner of the
square?
+q
P
d
-q
+q
d
A)
B)
Calculate E at point P.
D) know d
E) Need to
know d & q

Qi
E   k 2 rˆi
ri
i
1  q
Ex 
2

4 o  d
1  q
Ey 
2

4 o  d
27
Need to
C) E  0
 
cos
2
4 
2d
q
( )
q
( 2d )
2
sin
 
4 
Electricity & Magnetism Lecture 2, Slide 10
Continuous Charge Distributions
“I don't understand the whole dq thing and lambda.”
Summation becomes an integral (be careful with vector nature)

Qi
E   k 2 rˆi
ri
i

dq
E   k 2 rˆ
r
WHAT DOES THIS MEAN ?
Integrate over all charges (dq)
r is vector from dq to the point at which E is defined
Linear Example:
l  Q/L
dE
pt for E
r
charges
30
dq  l dx
Electricity & Magnetism Lecture 2, Slide 11
Charge Density
“What are the units for charge density (lambda)? .”
Some Geometry
Linear (l  Q/L) Coulombs/meter
Surface (s  Q/A) Coulombs/meter2
Volume (r  Q/V) Coulombs/meter3
Asphere  4R 2
Acylinder  2RL
Vsphere  43 R 3
Vcylinder  R 2 L
What has more net charge?.
A) A sphere w/ radius 2 meters and volume charge density r = 2 C/m3
B) A sphere w/ radius 2 meters and surface charge density s = 2 C/m2
C) Both A) and B) have the same net charge.
QA  rV  r 43 R3
QB  sA  s 4R 2
33
QA r 43 R 3 1 r


R
QB s 4R 2 3 s
Electricity & Magnetism Lecture 2, Slide 12
CheckPoint
10)
A) (EA<EB) “Electric Field at point A cancels out
to be zero and electric field at point B
experiences E field from both line to move
upward.”
C) (EA>EB) “A gets more of the field because it
is close to both lines of charge. B gets less of
the field because it is close to one and far away
from the other.”
37
Electricity & Magnetism Lecture 2, Slide 13
Calculation
y
“How is the integration of dE
over L worked out, step by step?”
P
r
h
dq  l dx
Charge is uniformly distributed along
the x-axis from the origin to x  a.
The charge density is l C/m. What is
the x-component of the electric field
at point P: (x,y)  (a,h)?
We know:
dq
What is r 2 ?
A) dx
x2
40
B)
x
x
a

dq
E   k 2 rˆ
r
dx
a 2 + h2
C)
ldx
a +h
2
2
D)
ldx
(a - x) 2 + h 2
E)
ldx
x2
Electricity & Magnetism Lecture 2, Slide 14
Calculation
Charge is uniformly distributed along
the x-axis from the origin to x  a.
The charge density is l C/m. What is
the x-component of the electric field
at point P: (x,y)  (a,h)?
dE
y
r
1
x

dq
E   k 2 rˆ
r
2
P
dEx
h
2
x
a
dq  l dx
We know:
dq
ldx

r 2 (a - x) 2 + h 2
We want:
Ex   dEx
What is dEx ?
A) dE cos 1
42
B) dE cos  2
C) dE sin 1
D) dE sin  2
Electricity & Magnetism Lecture 2, Slide 15
Calculation
dE
Charge is uniformly distributed along
the x-axis from the origin to x  a.
The charge density is l C/m. What is
the x-component of the electric field
at point P: (x,y)  (a,h)?
y
r
1

dq
E   k 2 rˆ
r
dE x
h
2
x
We know:
2
P
x
a
dq  l dx
dq
ldx

r 2 (a - x) 2 + h 2
Ex   dEx   dE cos  2
What is E x ?
kl cos  2 dx
0 (a - x) 2 + h 2
a
A)
B)
dx
2
2
(
a
x
)
+
h
0
lk cos  2 
C) A and B are both OK
45
a
cos2 DEPENDS ON x!
Electricity & Magnetism Lecture 2, Slide 16
Calculation
dE
Charge is uniformly distributed along
the x-axis from the origin to x  a.
The charge density is l C/m. What is
the x-component of the electric field
at point P: (x,y)  (a,h)?
y
r
1
dE x
h
2
x

dq
E   k 2 rˆ
r
2
P
x
a
dq  l dx
We know:
dq
ldx

r 2 (a - x) 2 + h 2
Ex   dEx   dE cos 2
What is cos 2 ?
A)
47
x
a +h
2
2
B)
a-x
(a - x) + h
2
2
C)
aa
2
2
aa2 ++ hh2
D)
aa
((aa--xx)2) 2+ +h 2h 2
Electricity & Magnetism Lecture 2, Slide 17
Calculation
Charge is uniformly distributed along
the x-axis from the origin to x  a.
The charge density is l C/m. What is
the x-component of the electric field
at point P: (x,y)  (a,h)?
We know:
y
2
P
r
1

dq
E   k 2 rˆ
r
dq
ldx

r 2 (a - x) 2 + h 2
dE
dE
x
x
dE x
h
2
x
a
dq  l dx
Ex   dEx   dE cos 2
cos  2 
a-x
(a - x) 2 + h 2
What is Ex (P) ?
a
Ex ( P)  lk  dx
0
49
a-x
((a - x)
2
+ h2
)
3/ 2
lk 
h
1 E x ( P) 
h 
h2 + a 2




Electricity & Magnetism Lecture 2, Slide 18
Homework Problem
50
Electricity & Magnetism Lecture 2, Slide 19