patterns of fields in space
DESCRIPTION
Patterns of Fields in Space. Box versus open surface. …no clue…. Seem to be able to tell if there are charges inside. Gauss’s law: If we know the field distribution on closed surface we can tell what is inside. Define Electric Flux. Gauss’s Law. Apply it. Symmetry. - PowerPoint PPT PresentationTRANSCRIPT
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
Define Electric Flux Gauss’s Law Apply it
Symmetry
Need a way to quantify pattern of electric field on surface: electric flux
1. Directionflux>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
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. Dependence on sum of charges inside4. Charges outside contribute zero
Gauss’s Law
0
ˆ
inside
surface
qAnE
204
1rQE
surface
AnrrQ ˆˆ
41
20
surface
ArQ
204
1
0
22
0
44
1
QrrQ
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
1rQE
2
1~r
E
2~ rA
2
1~r
E 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
The flux through the inner sphere is the same as the flux through the outer.
3. Gauss’s Law: The Shape of the Surface
A2 / A1 r22 / r1
2
E2A2 / E1A1 1
A2 R2 (r2 tan)2 r22
∆ 𝐴1⊥∝𝑟12
0
ˆ
inside
surface
qAnE
surfacesurface
AEAnE ˆ
2~ rA
2
1~r
E 2211 EAEA –
Outside charges contribute 0 to total flux
4. Gauss’s Law: Outside Charges
0
11 ˆ
QAnE
surface
0
22 ˆ
QAnE
surface
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
1rQE
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 204
1rQE
B. Inner Dashed Sphere:
0
2 04
rE 0E
The Electric Field of a Uniform Spherical Shell of Charge
Finally!
𝑟𝑟
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
Next Class is a Review