Answer questions Hour 1: Review: Electric Fields Charge ......Electric Field Lines 1. Direction of...
Transcript of Answer questions Hour 1: Review: Electric Fields Charge ......Electric Field Lines 1. Direction of...
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1P02 -
Class 02: Outline
Answer questionsHour 1:
Review: Electric FieldsChargeDipoles
Hour 2:Continuous Charge Distributions
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2P02 -
Last Time: FieldsGravitational & Electric
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3P02 -
Gravitational & Electric Fields
Mass M Charge q (±)
2ˆMG
r= −g r 2
ˆeqkr
=E rCREATE:
g m=F g E q=F E
This is easiest way to picture field
FEEL:
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4P02 -
PRS Questions:Electric Field
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5P02 -
Electric Field Lines1. Direction of field line at any point is tangent to
field at that point2. Field lines point away from positive charges
and terminate on negative charges3. Field lines never cross each other
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6P02 -
In-Class Problem
d
s
q− q+
P
ij
Consider two point charges of equal magnitude but opposite signs, separated by a distance d. Point Plies along the perpendicular bisector of the line joining the charges, a distance s above that line. What is the E field at P?
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7P02 -
Two PRS Questions:E Field of Finite Number of Point
Charges
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8P02 -
Charging
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9P02 -
How Do You Charge Objects?
• Friction• Transfer (touching)• Induction
Neutral----
++++
+q
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10P02 -
Demonstrations:Instruments for
Charging
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11P02 -
Electric Dipoles
A Special Charge Distribution
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12P02 -
Electric DipoleTwo equal but opposite charges +q and –q,
separated by a distance 2a
q
-q
2a charge×displacementˆ ˆ×2 2q a qa
≡
= =
p
j j
Dipole Moment
p
p points from negative to positive charge
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13P02 -
Why Dipoles?
Nature Likes To Make Dipoles!
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/20-Molecules2d/20-mole2d320.html
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14P02 -
Dipoles make Fields
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15P02 -
Electric Field Created by DipoleThou shalt use components!
3 3x ex xE k q
r r+ −
⎛ ⎞∆ ∆= −⎜ ⎟
⎝ ⎠
3 3y ey yE k q
r r+ −
+ −
⎛ ⎞∆ ∆= −⎜ ⎟
⎝ ⎠
3/2 3/22 2 2 2( ) ( )e
x xk qx y a x y a
⎛ ⎞⎜ ⎟= −⎜ ⎟⎡ ⎤ ⎡ ⎤+ − + +⎣ ⎦ ⎣ ⎦⎝ ⎠
3/2 3/22 2 2 2( ) ( )e
y a y ak qx y a x y a
⎛ ⎞− +⎜ ⎟= −⎜ ⎟⎡ ⎤ ⎡ ⎤+ − + +⎣ ⎦ ⎣ ⎦⎝ ⎠
2 3 3 3
ˆ ˆ ˆx yr r r r
∆ ∆= = +
r r i j
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16P02 -
PRS Question:Dipole Fall-Off
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17P02 -
Point Dipole Approximation
Take the limit r a>>
You can show…Finite Dipole
30
3 sin cos4x
pEr
θ θπε
→
( )23
0
3cos 14y
pEr
θπε
→ −Point Dipole
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18P02 -
Shockwave for Dipole
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/06-DipoleField3d/06-dipField320.html
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19P02 -
Dipoles feel Fields
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20P02 -
Demonstration:Dipole in Field
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21P02 -
Dipole in Uniform FieldˆE=E i
ˆ ˆ2 (cos sin )qa= +p i jθ θ
( ) 0net q q+ −= + = + − =F F F E E
tends to align with the electric field p
Total Net Force:
Torque on Dipole: = ×τ r F( )( )2 sin( )a qE θ=
= ×p Esin( )rFτ θ+= sin( )pE θ=
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22P02 -
Torque on DipoleTotal Field (dipole + background) shows torque:
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electr
ostatics/43-torqueondipolee/43-
torqueondipolee320.html
• Field lines transmit tension• Connection between dipole field and
constant field “pulls” dipole into alignment
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23P02 -
PRS Question:Dipole in Non-Uniform Field
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24P02 -
Continuous Charge Distributions
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25P02 -
Continuous Charge Distributions
( ) ?P =E
Vi
iQ q
Break distribution into parts:
= ∆∑
2ˆe
qkr∆
∆ =E r
E field at P due to ∆q
Superposition:
= ∆∑E E
V
dq→ ∫
d→ ∫ E
2ˆe
dqd kr
→ =E r
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26P02 -
Continuous Sources: Charge DensitydVdQ ρ=
R
L
2Volume V R Lπ= = QV
ρ =
LQ
=λ
QA
σ =
dAdQ σ=w
L
Area A wL= =
dLdQ λ=Length L=
L
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27P02 -
Examples of Continuous Sources: Line of charge
LQ
=λ
Length L=
L
dLdQ λ=
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/elect
rostatics/07-LineIntegration/07-
LineInt320.html
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28P02 -
Examples of Continuous Sources: Line of charge
LQ
=λ
Length L=
L
dLdQ λ=
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/elect
rostatics/08-LineField/08-
LineField320.html
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29P02 -
Examples of Continuous Sources: Ring of Charge
2Q
Rλ
π=dLdQ λ=
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/elect
rostatics/09-RingIntegration/09-
ringInt320.html
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30P02 -
Examples of Continuous Sources: Ring of Charge
2Q
Rλ
π=dLdQ λ=
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/elect
rostatics/10-RingField/10-
ringField320.html
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31P02 -
Example: Ring of Charge
P on axis of ring of charge, x from centerRadius a, charge density λ.
Find E at P
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32P02 -
Ring of Charge
Symmetry!0E⊥ =1) Think about it
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/elect
rostatics/09-RingIntegration/09-
ringInt320.html
2) Define Variables
dq dlλ=22 xar +=
( )a dλ ϕ=
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33P02 -
Ring of Charge3) Write Equation dq a dλ ϕ=
2
ˆe
rd k dqr
=E
a) My way
3x exdE k dqr
=
b) Another way
22 xar +=
3erk dqr
=
cos( )xdE d θ= E 2 3
1e e
x xk dq k dqr r r
= ⋅ =
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34P02 -
Ring of Charge4) Integrate
3x x exE dE k dqr
= =∫ ∫22 xar +=
dq a dλ ϕ=
3exk dqr
= ∫
Very special case: everything except dq is constant
2aλ π=dq∫2 2
0 0a d a d
π πλ ϕ λ ϕ= =∫ ∫
Q=
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35P02 -
Ring of Charge5) Clean Up
3x exE k Qr
=
( )3/ 22 2x exE k Q
a x=
+
0a →
( )3/ 22 2ˆ
exk Q
a x=
+E i
6) Check Limit
( )3/ 2 22
ex e
k QxE k Qxx
→ =
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36P02 -
r
2L
−2L
+
s
P
j
i
In-Class: Line of Charge
Point P lies on perpendicular bisector of uniformly charged line of length L, a distance s away. The charge on the line is Q. What is E at P?
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37P02 -
r
θ
θ
2L
−2L
+
xd ′
x′
xddq ′= λs
22 xsr ′+=
P
j
i
Hint: http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/07-LineIntegration/07-LineInt320.html
Typically give the integration variable (x’) a “primed” variable name.
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38P02 -
E Field from Line of Charge
2 2 1/ 2ˆ
( / 4)eQk
s s L=
+E j
Limits:
2ˆlim e
s L
Qks>>
→E j Point charge
ˆ ˆ2 2lim e es L
Qk kLs s
λ<<
→ =E j j Infinite charged line
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39P02 -
In-Class: Uniformly Charged Disk
P on axis of disk of charge, x from centerRadius R, charge density σ.
Find E at P
( 0 )x >
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40P02 -
Disk: Two Important Limits
( )1/ 22 2ˆ1
2disko
x
x R
⎡ ⎤⎢ ⎥= −⎢ ⎥+⎣ ⎦
E iσε
Limits:
2
1 ˆlim 4diskx R o
Qx>>
→E iπε
*** Point charge
ˆlim 2diskx R o<<
→E iσε
Infinite charged plane
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41P02 -
E for Plane is Constant????
1) Dipole: E falls off like 1/r3
2) Point charge: E falls off like 1/r2
3) Line of charge:E falls off like 1/r4) Plane of charge: E constant