divide shell into rings of charge, each delimited by the angle and the angle use polar coordinates...
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E 1
40
(Q)d
(d 2 (Rsin )2 )3/2
E 1
40
Q
2
(r Rcos)
(r Rcos )2 (Rsin)2 3/2
0
sind
Divide shell into rings of charge, each delimited by the angle and the angle +Use polar coordinates (r, ,).Distance from center: d=(r-Rcos)Surface area of ring:
2 (Rsin)R
Q Q2RsinR
4R2
Integration
R
R
Rcos
Rsin
r
d
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Chapter 22
Patterns of Fields in Space
• Electric flux• Gauss’s law
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What is in the box?
no charges? vertical charged plate?
Patterns of Fields in Space
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Box versus open surface
Seem to be able to tellif there are charges inside
…no clue…
Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.
Patterns of Fields in Space
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Need a way to quantify pattern of electric field on surface: electric flux
1. Direction
flux>0 : electric field comes outflux<0 : electric field goes in
+1 -10Relate flux to the angle between outward-going normal and E:
flux ~ cos()
Electric Flux: Direction of E
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2. Magnitude
flux ~ E
flux ~ Ecos()
Electric Flux: Magnitude of E
𝑓𝑙𝑢𝑥 𝐸 ∙ ��
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3. Surface area
flux through small area:
AnEflux ˆ~
Definition of electric flux on a surface:
surface
AnE ˆ
Electric Flux: Surface Area
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Perpendicular field
cosˆ AEAnE
AEAnE ˆ
Perpendicular area
coscosˆ yxEAEAnE
x y
AEAnE ˆ
Electric Flux: Perpendicular Field or Area
q
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surface
AnE ˆ
dAnE ˆ
Ad
AdE
AdE
surface closed a on flux electric
Adding up the Flux
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0
ˆ
inside
surface
qAnE
0
ˆ
insideqdAnE
Features:1. Proportionality constant2. Size and shape independence3. Independence on number of charges inside4. Charges outside contribute zero
Gauss’s Law
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0
ˆ
inside
surface
qAnE
204
1
r
QE
surface
Anrr
Qˆˆ
4
12
0
surface
Ar
Q2
04
1
0
22
0
44
1
Q
rr
Q
What if charge is negative?
Works at least for one charge and spherical surface
1. Gauss’s Law: Proportionality Constant
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0
ˆ
inside
surface
qAnE
204
1
r
QE
2
1~
rE
2~ rA
2
1~
rE universe would be
much different ifexponent was not exactly 2!
2. Gauss’s Law: The Size of the Surface
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0
ˆ
inside
surface
qAnE
E nA
surface EA
surface
All elements of the outer surface can be projected onto corresponding areas on the inner sphere with the same flux
3. Gauss’s Law: The Shape of the Surface
A2 / A1 r22 / r1
2
E2A2 / E1A1 1
A2 R2 (r2 tan)2 r22
∆ 𝐴1⊥=𝜋 𝑟12
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0
ˆ
inside
surface
qAnE
surfacesurface
AEAnE ˆ
2~ rA
2
1~
rE 2211 EAEA –
Outside charges contribute 0 to total flux
4. Gauss’s Law: Outside Charges
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0
11 ˆ
Q
AnEsurface
0
22 ˆ
Q
AnEsurface
0ˆ3 surface
AnE
0
ˆ
inside
surface
qAnE
5. Gauss’s Law: Superposition
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0
ˆ
inside
surface
qAnE
0
ˆ
insideqdAnE
Features:1. Proportionality constant2. Size and shape independence3. Independence on number of charges inside4. Charges outside contribute zero
Gauss’s Law and Coulomb’s Law?
204
1
r
QE
Can derive one from another
Gauss’s law is more universal:works at relativistic speeds
Gauss’s Law
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0
ˆ
inside
surface
qAnE
1. Knowing E can conclude what is inside2. Knowing charges inside can conclude what is E
Applications of Gauss’s Law
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Symmetry: Field must be perpendicular to surfaceEleft=Eright
0
ˆ
inside
surface
qAnE
2EAbox Q / A Abox
0
E Q / A 20
The Electric Field of a Large Plate
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Symmetry: 1. Field should be radial2. The same at every location
on spherical surface
0
ˆ
inside
surface
qAnE
A. Outer Dashed Sphere:
0
24
QrE 2
04
1
r
QE
B. Inner Dashed Sphere:
0
2 04
rE 0E
The Electric Field of a Uniform Spherical Shell of Charge
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0
ˆ
inside
surface
qAnE
Is Gauss’s law still valid?
Can we find E using Gauss’s law?
The Electric Field of a Uniform Cube
Without symmetry, Gauss’s law loses much of its power.
Yes, it’s always valid.
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Gauss’s Law for Electric Dipole
No symmetry
Direction and Magnitude of E varies
NumericalSolution
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Clicker Question
What is the net electric flux through the box?
A) 0 VmB) 0.36 VmC) 0.84 VmD) 8.04 VmE) 8.52 Vm
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Can we have excess charge inside a metal that is in static equilibrium?
Proof by contradiction:
0
ˆ
inside
surface
qAnE
=0
00
insideq
Gauss’s Law: Properties of Metal
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0
ˆ
inside
surface
qAnE
=0
00
insideq
Gauss’s Law: Hole in a Metal
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+5nC
0
ˆ
inside
surface
qAnE
=0
00
insideq
0 insidesurface qq
nC 5 surfaceq
Gauss’s Law: Charges Inside a Hole
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Review for Midterm