Comments: Yours and Ours
1. Please respect your fellow students - please operate only your own clicker
2. Quite a bit of homework this week and building will be closed Sun/Mon
3. Don’t panic!......Don’t be intimidated by integrals!
Physics 212 Lecture 2, Slide 1
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 quicklyI 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.
Electricity & Magnetism
Lecture 2
Today’s Concepts:A) The Electric Field
B) Continuous Charge Distributions
Electricity & Magnetism Lecture 2, Slide 2
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
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:
+q1 -> -q1 direction reversed
Coulomb’s Law (from last time)
F2,1
F3,1
F4,1
F1
q1
q2
q3
q4
F2,1
F3,1
F4,1
F1
F2,1
F3,1
F4,1
F1
F2,1
F3,1
F4,1
F1
q1
q2
q3
q4
142
14
41132
13
31122
12
21 ˆˆˆ rr
qkqr
r
qkqr
r
qkq++
Electricity & Magnetism Lecture 2, Slide 3
MATH:
1F
142
14
4132
13
3122
12
2
1
1 ˆˆˆ rr
kqr
r
kqr
r
kq
q
F++
E
09
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
E2
E3
E4
E
Field points toward negative andAway from positive charges.
“Can you explain the derivations of the equations for electric fields? ““What is the essence of an electric field? “
Electric Field
q4
q2
q3
Electricity & Magnetism Lecture 2, Slide 4
q
FE
rr
QkE ˆ
2
i
i
i
i rr
QkE ˆ
2
12
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.
“I can't figure out the field directions at all.”
CheckPoint
Electricity & Magnetism Lecture 2, Slide 5
A
Bx
y
+Q -Q
What is direction at point A
a) Up b) down c) Left d) Right e) zero
What is direction at point B
a) Up b) down c) Left d) Right e) zero
15
E
E
CheckPoint
+Q
+Q
+Q
-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)
A
A
Case A Case B
Electricity & Magnetism Lecture 2, Slide 6A B Equal
“Same because
the forces are not
canceling each
other out” “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”
“in case B, the two charges
carry same charges, they will
counteract in the line parallel to
the line connecting them”
18
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.
x
y
q1 q2
(-a,0) (a,0)
(0,d)
Which of the following statements is true:
A) Both charges are negativeB) Both charges are positiveC) The charges are oppositeD) There is not enough information to tell how the charges are
related
Two Charges
E
Electricity & Magnetism Lecture 2, Slide 720
A B C 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.”
CheckPoint
Electricity & Magnetism Lecture 2, Slide 9
INTERESTING: statement is correct, but given in support of “to the left” !!
24
Electricity & Magnetism Lecture 2, Slide 10
Example
Calculate E at point P.
“Show me more electric field examples, please!”
d
P
d
A) B) C) D)Need to know d
Need to know d & q
E)
What is the direction of the electric field at point P, the unoccupied corner of the square?
0E
( )
-
4cos
24
122
d
q
d
qE
o
x
( )
-
4sin
24
122
d
q
d
qE
o
y
-q +q
+q
i
i
i
i rr
QkE ˆ
2
27
l Q/L
Summation becomes an integral (be careful with vector nature)
“I don't understand the whole dq thing and lambda.”
WHAT DOES THIS MEAN ?
Integrate over all charges (dq)
r is vector from dq to the point at which E is defined
r
dE
Continuous Charge Distributions
Linear Example:
charges
pt for E
dq l dx
Electricity & Magnetism Lecture 2, Slide 11
i
i
i
i rr
QkE ˆ
2
r
r
dqkE ˆ
2
30
Charge Density
Linear (l Q/L) Coulombs/meter
Surface (s Q/A) Coulombs/meter2
Volume (r Q/V) Coulombs/meter3
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.
“What are the units for charge density (lambda)? .”
Some Geometry
24 RAsphere
3
34 RVsphere LRVcylinder
2
RLAcylinder 2
3
34 RVQA rr
24 RAQB ss
RR
R
Q
Q
B
A
s
r
s
r
3
1
4 2
3
34
Electricity & Magnetism Lecture 2, Slide 1233
10)
CheckPoint
Electricity & Magnetism Lecture 2, Slide 13
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
“How is the integration of dE
over L worked out, step by step?”
Calculation
2x
dxA) B) C) D) E)
What is ? 2r
dq
22 ha
dx
+22 ha
dx
+
l22)( hxa
dx
+-
l2x
dxl
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)?
x
y
a
h
P
x
r
dq l dx
Electricity & Magnetism Lecture 2, Slide 14
We know:
rr
dqkE ˆ
2
40
Calculation
We know:
222 )( hxa
dx
r
dq
+-
l
xx dEE
What is ?
A) B) C) D)
xdE
1cosdE2cosdE
1sindE 2sindE
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)?
xa
P
x
r
1 2
2
dq l dx
h
xdE
dE
y
Electricity & Magnetism Lecture 2, Slide 15
rr
dqkE ˆ
2 We want:
42
We know:
cos2 DEPENDS ON x!
Calculation
2cosdEdEE xx222 )( hxa
dx
r
dq
+-
l
What is ?
A) B)
C) A and B are both OK
xE
+-
a
hxa
dxk
0
222)(
cosl +-
a
hxa
dxk
0
22
2
)(
cosl
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)?
xa
P
x
r
1 2
2
dq l dx
h
xdE
dE
y
Electricity & Magnetism Lecture 2, Slide 16
rr
dqkE ˆ
2
45
We know:
Calculation
2cosdEdEE xx222 )( hxa
dx
r
dq
+-
l
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)?
xa
P
x
r
1 2
2
dq l dx
h
xdE
dE
y
Electricity & Magnetism Lecture 2, Slide 17
rr
dqkE ˆ
2
22 ha
x
+
What is ?
A) B) C) 22 ha
a
+ 22)( hxa
a
+-D)
22 ha
a
+22)( hxa
a
+-22)( hxa
xa
+-
-
2cos
47
We know:
xdE
Calculation
222 )( hxa
dx
r
dq
+-
l 2cosdEdEE xx
222
)(cos
hxa
xa
+-
-
( ) 2/3220 )(
)(hxa
xadxkPE
a
x
+-
- l
What is ? )(PEx
+-
221)(
ah
h
h
kPEx
l
xa
P
x
r
1 2
2
dq l dx
h
xdE
dE
y
Electricity & Magnetism Lecture 2, Slide 18
rr
dqkE ˆ
2
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)?
49