Physics 2220 Chapter 24 Gausss Law

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Physics 2220Chapter 24 Gausss Law

• Quiz Given that the contribution to the
  electric field on the axis of an annulus of
  radius R and infinitesimal thickness dR is dEz
  2pske z RdR z2 R2 3/2 where s is the
  surface charge density, show that the electric
  field a distance z above an infinite plane of
  charge is Ez s/2eo, where ke 1/4peo.

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Physics 2220Chapter 24 Gausss Law

• Geometry

dEz 2pskezRdR z2 R2 3/2
dEz
z
dA 2pRdR
r z2 R2 1/2
R
s
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Physics 2220Chapter 24 Gausss Law

• Calculus

dEz 2pskezRdR z2 R2 3/2
dEz
Ez s/2eo
z
s
Ez ?08 2pskezRdR z2 R2 3/2
Ez 2pskez (-1)z2 R2 1/208
Ez 2pskez1/z - 0 2pske
Ez 2ps/4peo s/2eo
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Physics 2220 Chapter 24 Gausss Law

• 1' Lesson
• The Electric Flux is defined to be the inner
  product (dot product) of the surface element and
  the electric field passing through a surface.
• Gausss Law states that the integrated flux
  through a closed surface is equal to the enclosed
  charge divided by the permittivity of free
  space.
• Gausss Law is a powerful tool in determining
  the electric field for many different charge
  distributions.

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Physics 2220Chapter 24 Gausss Law

• Electric Flux

• A useful abstraction FE

dA
dFE E?dA
dFE E cos ? dA
E
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Physics 2220Chapter 24 Gausss Law

• Gausss Law

• The total flux through a closed Gaussian
  surface is equal to the charge enclosed divided
  by the permittivity of free space.

FE ?S E?dA
dA
E
Gausss Law
q
? E?dA q/eo
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Physics 2220Chapter 24 Gausss Law

• Karl Friedrich Gauss
• (1777-1855)
• One of all-time greatest mathematicians
• German child prodigy
• Proved Fundamental Theorem of Algebra
• Contributed to E-M

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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• An infinite sheet of charge

E
dA
dFE 2Ez dA
dq s dA
2Ez dA s dA/ eo
Ez s / 2eo
s
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Physics 2220Chapter 24 Gausss Law

• Electric Field inside a Conductor

e
e
Equilibrium E 0
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Physics 2220Chapter 24 Gausss Law

• Electric Field inside a Conductor

E
dA
s
E 0
Ez dA sdA/ eo
Ez s/ eo
s
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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• An infinite sheet of charge

E
dA
dFE 2Ez dA
dq s dA
2Ez dA s dA/ eo
Ez s / 2eo
s
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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• Calculate the electric field at a distance R
  from an infinite line of charge with a uniform
  charge density ?.

R
?
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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• Calculate the electric field at a distance R
  from an infinite line of charge with a uniform
  charge density ?.

ER (2pR) L ? L/eo
L
ER ?/(2peo R)
R
?
ER
dA
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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• A spherically symmetric charge density ?

ER (4pR2 ) (1/eo )?? dV
?
ER 1/(4 peoR2 )?? dV
ER
R
dA
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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• A spherically symmetric charge density ?

ER (4pR2 ) (1/eo )?? dV
?
ER 1/(4 peoR2 )?? dV
R
ER q/(4 peoR2 ) Rgta
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Physics 2220Chapter 24 Gausss Law

• Why does Gausss Law work?

Solid angle ?O ?A?r / R2 O 4pR2/R2 4p For
point charge ?FE ?A?rER ?O ER R2 But ER
q/4peoR2 FE O ER R2 4p (q/4peoR2 )R2
q/eo
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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• What is the electric field inside a uniformly
  charged spherical shell of charge?

ER (4pR2 ) (1/eo )?? dV
ER 1/(4 peoR2 )?? dV
ER
R
dA
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Physics 2220Chapter 24 Gausss Law

• Applications of Gausss Law
• What is the charge distribution if the electric
  field inside a conductor is zero?

ER (4pR2 ) (1/eo )?? dV
ER 1/(4 peoR2 )?? dV
ER
R
dA
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Physics 2220Chapter 24 Gausss Law

• Summary
• FE ?S E?dA Electric Flux
• ? E?dA q/eo Gausss Law
• Ez s / eo E outside conducting plate E 0
  inside conductor
• Ez s / 2eo E for infinite sheet of s
• ER q/(4 peoR2 ) E outside spherical q
• ER ?/(2peo R) E for line charge of ?