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Chapter 23: 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 A

E A

A

E E A

E 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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E

q

rrq

kdArq

k

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=d

constant

1

ddAEdA

dAEAdE

rdAr

E

Adding up the flux over the surface of one of the spheres

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q

dA

E

d E dA

EdAE

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

0enclosedq

AdE

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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 q

qAdEE

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

rq

E

qrEEAAdE

R r

E

rr=R

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Field of a uniform ball of charge, with total charge q and radius R

Spherical symmetry => spherical Gaussian surfaces

E constant on surface, E perpendicular to surface

exterior:

interior:

20

0

2

4

4

rq

E

qrEEAAdE

R r

E

rr=R

30

20

3

3

20

3

3

3

3

0

2

444

3434

4

Rqr

rRr

q

r

qE

Rr

qqRr

q

q

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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0enclosedq

AdE

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

lx x

E

AE

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0enclosedq

AdE

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