PHY2049: Chapter 23 1
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Gauss' Law and Electric Fields: Calculation and Symmetries, Study notes of Physics
A chapter from a physics textbook focusing on gauss' law and its application to calculate electric fields in various symmetrical charge distributions, including spherical, cylindrical, and rectangular symmetries. It covers topics such as conductors, spherical symmetry, axial symmetry, and rectangular symmetry, providing equations and solutions for different scenarios.
Typology: Study notes
Pre 2010
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PHY2049: Chapter 23
First Beams at LHC!
PHY2049: Chapter 23
Gauss’ Law Says
Î
Total flux depends only on the amount ofenclosed charge, not on shape and size of theGaussian surface.
Î
Charges outside have
no effect on the total
flux. ÎThis does not mean they do not contribute to E
Remember:
Outward E field, flux > 0 Inward E field,
flux < 0
enc ε^0 q
d S^
∫^
A
E
PHY2049: Chapter 23
Use Gauss’ Law to Calculate E Fields Î
Spherical symmetry^
E field vs r inside uniformly charged sphere
Charges on concentric spherical conducting shells Î
Cylindrical symmetry^
E field vs r for line charge
E field vs r inside uniformly charged cylinder Î
Rectangular symmetry^
E field for charged plane
E field between conductors, e.g. capacitors
PHY2049: Chapter 23
Spherical Symmetry (1)
ÎInsulator or conductor?
^ Conducting sphere cannot be uniformly charged ÎInside
^ By symmetry, E must be radially symmetric ^ E field has constant mag.,
⊥^
to Gaussian surface
ÎOutside
Gaussian surface
(sphere)
Gauss’ Law
Solve for E
Uniformly Charged Sphere
enc^0
2 ) (^4) (
q ε
rπ E
d S^
∫^
A
E
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3 4 3 Q Rπ Q ρ = 3 3
3 ) (^4) ( 3
r R Q rπ ρ^
= Q 3 0 2 0
3 3
(^14)
4
/
Qr^ R πε
r πε
R Qr E^
=
=
2 0 (^14)
Q r πε
E^
=
Volume chargedensity
PHY2049: Chapter 23
Axial Symmetry (1):
Line Charge
ÎInfinitely long line, uniformly charged
^ By symmetry, E must be axially symmetric ^ On curved surface, E field has constantmag.,
⊥
to Gaussian surface
^ Through top and bottom surfaces, no
Φ
E
since E is ||
Solve for E
enc^0
0 0 ) 2 (^
q ε
rπ h E
d S^
∫^
A
E
Gauss’ Law
hλ
λ r πε E
0 (^12) =^
λ: linear charge density
PHY2049: Chapter 23
Axial Symmetry (2):
Uniformly Charged Cylinder
ÎInfinitely tall cylinder, uniformly charged
^ By symmetry, E must be axially symmetric ^ On curved surface, E field has constant mag.,^ ⊥
to Gaussian surface ^ Through top and bottom surfaces, no
Φ
sinceE
E is ||
)
(^
(^2) r πh ρ
Solve for E
enc^0
0 0 ) 2 (^
q ε
rπ h E
d S^
∫^
A
E
Gaussian surface
(cylindrical) ρ
R r
h
Gauss’ law
rρ ε
E
0 (^12) =
ρ: volumecharge density
PHY2049: Chapter 23
λ πε
E
0 2
1
Some Comparisons
total^2 r 0
q
E^
r
0 σ 2 ε
E^
No distance dependence!
Î
Spherically symmetric charge distribution Î
Uniformly charged, infinitely long (i.e., verylong) line Î
Uniformly charged plane