Divide shell into rings of charge, each delimited by the angle and the angle Use polar coordinates...
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Transcript of Divide shell into rings of charge, each delimited by the angle and the angle Use polar coordinates...
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
Chapter 22
Patterns of Fields in Space
• Electric flux• Gauss’s law
What is in the box?
no charges? vertical charged plate?
Patterns of Fields in Space
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
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
2. Magnitude
flux ~ E
flux ~ Ecos()
Electric Flux: Magnitude of E
𝑓𝑙𝑢𝑥 𝐸 ∙ ��
3. Surface area
flux through small area:
AnEflux ˆ~
Definition of electric flux on a surface:
surface
AnE ˆ
Electric Flux: Surface Area
Perpendicular field
cosˆ AEAnE
AEAnE ˆ
Perpendicular area
coscosˆ yxEAEAnE
x y
AEAnE ˆ
Electric Flux: Perpendicular Field or Area
q
surface
AnE ˆ
dAnE ˆ
Ad
AdE
AdE
surface closed a on flux electric
Adding up the Flux
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
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
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
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
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
0
11 ˆ
Q
AnEsurface
0
22 ˆ
Q
AnEsurface
0ˆ3 surface
AnE
0
ˆ
inside
surface
qAnE
5. Gauss’s Law: Superposition
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
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
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
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
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.
Gauss’s Law for Electric Dipole
No symmetry
Direction and Magnitude of E varies
NumericalSolution
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
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
0
ˆ
inside
surface
qAnE
=0
00
insideq
Gauss’s Law: Hole in a Metal
+5nC
0
ˆ
inside
surface
qAnE
=0
00
insideq
0 insidesurface qq
nC 5 surfaceq
Gauss’s Law: Charges Inside a Hole
Review for Midterm