p212c22: 1 Chapter 22: Gauss’s Law Gauss’s Law is an alternative formulation of the relation between an electric field and the sources of that field in terms of electric flux. Electric Flux E through an area A ~ Number of Field Lines which pierce the area depends upon geometry (orientation and size of area, direction of E) electric field strength (| E| ~ density of field lines) E E A EA A E E A EA cos
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p212c22: 1
Chapter 22: Gauss’s LawGauss’s Law is an alternative formulation of the relation between an electric field and the sources of that field in terms of electric flux.
Electric Flux E through an area A~ Number of Field Lines which pierce the areadepends upon
geometry (orientation and size of area, direction of E)electric field strength (|E| ~ density of field lines)
E E AE A
A
E E AE A
cos
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Gauss’s law relates to total electric flux through a closed surface to the total enclosed charge.
Start with single point charge enclosed within an arbitrary closed surface.
Add up all contributions d.
E E dA
q
AdEd E
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o
E
q
rrqkdA
rqk
AdE
2
22 4
intermediate steps: charge at the center of a spherical surfacetwo patches of area subtending the same solid angle
q
222111
22
E=dconstant
1
ddAEdAdAEAdE
rdAr
E
Adding up the flux over the surface of one of the spheres
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q
dA
E
d E dAEdA
E
sphere
cos
For a charge in an arbitrary surfaceProject area increment onto “nearest sphere”: Flux through area = flux through area increment on “nearest sphere” with same solid angle.
Flux through “nearest sphere” area increment = flux through area increment on a common sphere for same solid angle.
Add up over all solid angles => over entire surface of common sphere => simple sphere results.
oE
q
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For charges located outside the closed surface
number of field lines exiting the surface (E) = number of field lines entering the surface (E)
=> no net contribution to E
Gauss’s Law:
o
enclosedE
QAdE
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Using Gauss’s Law•Select the mathematical surface (a.k.a. Gaussian Surface)
-to determine the field at a particular point, that point must lie on the surface-Gaussian surface need not be a real physical surface in empty space, partially or totally embedded in a solid body
•Gaussian surface should have the same symmetries as charge distribution.
-concentric sphere, coaxial cylinder, etc.
•Closed Gaussian surface can be thought of as several separate areas over which the integral is (relatively) easy to evaluate.
-e.g. coaxial cylinder = cylinder walls + caps
•If E is perpendicular to the surface (E parallel to dA) and has constant magnitude then
•If E is tangent (parallel) to the surface (E perpendicular to dA) then
EAAdE
0 AdE
0enclosedqAdE
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Conductors and Electric Fields in ElectrostaticsConductors contain charges which are free to moveElectrostatics: no charges are movingF = q E
=> for a conductor under static conditions, the electric field within the conductor is zero. E = 0
For any point within a conductor, and all Gaussian surfaces completely imbedded within the conductor
0000
enclosedenclosed qqAdEE
q = 0 within bulk conductor
=> all (excess) charge lies on the surface! (for a conductor under static conditions)
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Faraday “ice-pail” experimentcharged conducting ball lowered to interior of “ice-pail”ball touches pail => part of interior of conductor
Conductor with void: all charge lies on outer surface unless there is an isolated charge within void.
Ball comes out uncharged => verifies Gauss’s Law => Coulomb’s Law
Modern versions establish exponent in Coulomb’s = 2 to 16 decimal places
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Field of a conducting sphere, with total charge q and radius R
Spherical symmetry => spherical Gaussian surfaces
E constant on surface, E perpendicular to surface
E = 0 on interior
exterior:
20
0
2
4
4
rqE
qrEEAAdE
R r
E
rr=R
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Field of a uniform ball of charge, with total charge q and radius RSpherical symmetry => spherical Gaussian surfaces
E constant on surface, E perpendicular to surface
exterior:
interior:
20
0
2
4
4
rqE
qrEEAAdE
R r
E
rr=R
30
20
3
3
20
3
3
3
30
2
444
3434
4
Rqr
rRrq
rqE
Rrqq
Rr
qrEEAAdE
enclosed
enclosedenclosed
enclosed
r
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Line of charge (infinite), charge per unit length cylindrical symmetry, E is radially outward (for positive )
Gaussian surface: finite cylinder, length l and radius r
Caps: E parallel to surface, = 0
Cylinder: E perpendicular to the surface
0
00
2
200
2
rE
lq
rlEAdE
rElEAAdE
r
enclosed
r
l
r
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Symetry is the Key!
rkESymetrylCylindricar
kQESymetrySpherical
enc
enc
22
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0enclosedqAdE
Field of an infinite sheet of charge, charge per area infinite plane, E is perpendicular to the plane (for positive ) with reflection symmetry
Gaussian surface: finite cylinder, length 2x centered on plane, caps with area A
Tube: E parallel to surface, = 0
2 Caps: E perpendicular to the surfaces
0
00
2
20
x
enclosed
x
x
E
Aq
AEAdE
AEAdE
x x
E
AE
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0enclosedqAdE
Two oppositely charged infinite conducting plates ()planar geometry, E is perpendicular to the plane
Gaussian surfaces: finite cylinder, length l centered on plane, caps with area A
Tube: E parallel to surface, = 0
Caps: E perpendicular to the surfaces
0
00
00
x
enclosed
x
x
E
Aq
AEAdE
AEAdE
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