Class 04: Outline
Hour 1:
Working In Groups
Expt. 1: Visualizations
Hour 2:
Electric Potential
Pick up Group Assignment at Back of Room
P04 - 1
Groups
P04 - 2
Advantages of Groups
• Three heads are better than one
• Don’t know? Ask your teammates
• Do know? Teaching reinforces knowledge
Leave no teammate behind!
• Practice for real life – science and
engineering require teamwork; learn to
work with others
P04 -
What Groups Aren’t
• A Free Ride
We do much group based work (labs &
Friday problem solving). Each individual
must contribute and sign name to work
If you don’t contribute (e.g. aren’t in class)
you don’t get credit
P04 -
Group Isn’t Working Well?
1. Diagnose problem and solve it yourself
-- Most prevalent MIT problem: free rider.
2. Talk to Grad TA
3. Talk to the teamwork consultant
Don’t wait: Like most problems,
teamwork problems get worse
the longer you ignore them
P04 -
Introduce Yourselves
Please discuss:
•
•
•
•
•
What is your experience in E&M?
How do you see group working?
What do you expect/want from class?
What if someone doesn’t participate?
What if someone doesn’t come to class?
Try to articulate solutions to foreseeable
problems now (write them down)
P04 -
Experiment 1: Visualizations
Need experiment write-up from course
packet.
Turn in tear sheet at end of class
Each GROUP hands in ONE tear sheet
signed by each member of group
P04 - 7
Last Time:
Gravitational & Electric Fields
P04 - 8
Gravity - Electricity
CREATE:
FEEL:
Mass M
Charge q (±)
G
M
g = −G 2 rˆ
r
G
q
E = ke 2 rˆ
r
G
G
Fg = mg
G
G
FE = qE
This is easiest way to picture field
P04 - 9
Potential Energy
and Potential
Start with Gravity
P04 -10
Gravity: Force and Work
Gravitational Force on m due to M:
G
Mm
Fg = −G 2 rˆ
r
Work done by gravity moving m from A to B:
Wg = ∫
B
A
G
G
Fg ⋅ d s
PATH
INTEGRAL
P04 -11
Work Done by Earth’s Gravity
Work done by gravity moving m from A to B:
G
G
Wg = ∫ Fg ⋅ d s
B
(
GMm ⎞
⎛
= ∫ ⎜ − 2 rˆ ⎟ ⋅ dr rˆ + rdθ θˆ
r
⎠
A⎝
)
rB
GMm ⎤
GMm
⎡
= ∫ − 2 dr =
r
⎢⎣ r ⎥⎦ r
rA
⎛ 1 1 ⎞
= GMm ⎜ − ⎟
⎝ rB rA ⎠
rB
A
What is the sign moving from rA to rB?
P04 -12
Work Near Earth’s Surface
G
GM
G roughly constant: g ≈ − 2 yˆ = − g yˆ
rE
Work done by gravity moving m from A to B:
G
B
G
G
Wg = ∫ Fg ⋅ d s = ∫ ( − mg yˆ ) ⋅ d s
A
yB
= − ∫ mgdy = − mg ( yB − y A )
yA
Wg depends only on endpoints
– not on path taken –
Conservative Force
P04 -13
Potential Energy (Joules)
∆ U g = U B − U A = −∫
B
A
G
G
Fg ⋅ d s = − Wg = + Wext
G
GMm
GMm
ˆ
(1) Fg = −
r
→ Ug = −
+U0
2
r
r
G
→ U g = mgy +U0
(2) Fg = − m g yˆ
• U0: constant depending on reference point
• Only potential difference ∆U has
physical significance
P04 -14
Gravitational Potential
(Joules/kilogram)
Define gravitational potential difference:
∆ Vg =
∆U g
m
= −∫
B
A
G
BG
G
G
(Fg / m) ⋅ d s = − ∫ g ⋅ d s
A
G
G
Just as Fg → gN , ∆ U g → ∆ Vg
N
N
N
Force
Field
Energy
Potential
That is, two particle interaction Æ single particle effect
P04 -15
PRS Question:
Masses in Potentials
P04 -16
Move to Electrostatics
P04 -17
Gravity - Electrostatics
Mass M
Charge q (±)
G
M
g = −G 2 rˆ
r
G
G
Fg = mg
G
q
E = ke 2 rˆ
r
G
G
FE = qE
Both forces are conservative, so…
∆ Vg = − ∫
∆ U g = −∫
B
A
B
A
G G
g⋅d s
G
G
Fg ⋅ d s
G G
∆ V = −∫ E ⋅ d s
A
B G
G
∆ U = − ∫ FE ⋅ d s
B
A
P04 -18
Potential & Energy
G G
∆ V ≡ −∫ E ⋅ d s
B
Units:
Joules/Coulomb
= Volts
A
Work done to move q from A to B:
Wext = ∆U = U B − U A
= q ∆V
Joules
P04 -19
Potential: Summary Thus Far
Charges CREATE PotentialG Landscapes
G
V (r ) = V0 + ∆ V ≡ V0 −
r
∫
G G
E⋅d s
"0"
P04 -20
Potential Landscape
Positive Charge
Negative Charge
P04 -21
Potential: Summary Thus Far
Charges CREATE PotentialG Landscapes
G
V (r ) = V0 + ∆ V ≡ V0 −
r
∫
G G
E⋅d s
"0"
Charges FEEL Potential Landscapes
G
G
U ( r ) = qV ( r )
We work with ∆U (∆V) because
only changes matter
P04 -22
Potential Landscape
Positive Charge
Negative Charge
P04 -23
3 PRS Questions:
Potential & Potential Energy
P04 -24
Creating Potentials:
Two Examples
P04 -25
Potential Created by Pt Charge
∆V = VB − VA = − ∫
B
A
G G
E ⋅ ds
B dr
rˆ G
= − ∫ kQ 2 ⋅ d s = − kQ ∫ 2
A
A r
r
⎛1 1⎞
= kQ ⎜ − ⎟
⎝ rB rA ⎠
B
Take V = 0 at r = ∞:
kQ
VPoint Charge (r ) =
r
G
rˆ
E = kQ 2
r
G
d s = dr rˆ + r dθ θˆ
P04 -26
2 PRS Questions:
Point Charge Potential
P04 -27
Potential Landscape
Positive Charge
Negative Charge
P04 -28
Deriving E from V
P04 -29
Deriving E from V
G G
∆ V = −∫ E ⋅ d s
B
A
A = (x,y,z), B=(x+∆x,y,z)
G
∆ s = ∆ x ˆi
( x +∆ x , y , z )
∆V = −
∫
G G
G
G G
E ⋅ d s ≅ −E ⋅ ∆ s = −E ⋅ (∆ x ˆi ) = − Ex ∆ x
( x, y, z )
∆V
∂ V Ex = Rate of change in V
Ex ≅ −
→−
∆x
∂ x with y and z held constant
P04 -30
Deriving E from V
If we do all coordinates:
G
⎛ ∂V ˆ ∂V ˆ ∂V ˆ ⎞
E = −⎜
i+
j+
k⎟
∂y
∂z ⎠
⎝ ∂x
⎛ ∂ ˆ ∂ ˆ ∂ ˆ⎞
= −⎜
i+
j + k ⎟V
∂y
∂z ⎠
⎝ ∂x
G
E = −∇ V
Gradient (del) operator:
∂ ˆ ∂ ˆ ∂ ˆ
j+ k
∇≡
i+
∂x
∂y
∂z
P04 -31
In Class Problem
From this plot
of potential vs.
position,
create a plot
of electric field
vs. position
Bonus: Is
there charge
somewhere?
Where?
P04 -32
Configuration Energy
P04 -33
Configuration Energy
How much energy to put two charges as pictured?
1) First charge is free
2) Second charge sees first:
U12 = W2 = q2V1 =
1
q1q2
4πε o r12
P04 -34
Configuration Energy
How much energy to put three charges as pictured?
1) Know how to do first two
2) Bring in third:
q3 ⎛ q1 q2 ⎞
+
W3 = q3 ( V1 + V2 ) =
⎜
⎟
4π ε 0 ⎝ r1 3 r2 3 ⎠
Total configuration energy:
1 ⎛ q1q2 q1q3 q2 q3
U = W2 + W3 =
+
+
⎜
r13
r23
4πε 0 ⎝ r12
⎞
⎟ = U12 + U13 + U 23
⎠
P04 -35
In Class Problem
What is the electric potential
in volts at point P?
How much energy in joules is
required to put the three
charges in the configuration
pictured if they start out at
infinity?
Suppose you move a fourth change +3Q from
infinity in to point P. How much energy does
that require (joules)?
P04 -36