physics 2113 lecture 12: wed 11 feb gauss’ law iii physics 2113 jonathan dowling carl friedrich...
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Physics 2113Physics 2113Lecture 12: WED 11 FEBLecture 12: WED 11 FEB
GaussGauss’’ Law III Law III
Physics 2113
Jonathan Dowling
Carl Friedrich Gauss1777 – 1855
Flux Capacitor (Operational)
What? The Flux!What? The Flux!
Constant Area:A
Changing Area:
dA
E
GaussGauss’’ Law: General Case Law: General Case
• Consider any ARBITRARY CLOSED surface S -- NOTE: this “Gaussian Surface” does NOT have to be a “real” physical object!
• The TOTAL ELECTRIC FLUX through S is proportional to the TOTAL CHARGE ENCLOSED!
• The results of a complicated integral is a very simple formula: it avoids long calculations! (One of Maxwell’s 4 equations!)
SE +q
ExamplesExamples
What is FluxThrough Surfaces:S1 = S2 = S3 = S4 =
+q/ε0
–q/ε0
00
What is total charge inside?
GaussGauss’’ Law: Cylindrical Symmetry Law: Cylindrical Symmetry
• Charge of q = 10 C is uniformly spread over a line of length L = 1 m.
• Use Gauss’ Law to compute magnitude of E at a perpendicular distance of 1 mm from the center of the line.
• Approximate as infinitely long line — E radiates outwards.
• Choose cylindrical surface of radius r, length L co-axial with line of charge.
r = 1 mmE = ?
L=1 m
Line of Charge: λ = q/L Units: [C/m]
GaussGauss’’ Law: Cylindrical Symmetry Law: Cylindrical Symmetry
• Approximate as infinitely long line — E radiates outwards.
• Choose cylindrical surface of radius r, length L co-axial with line of charge.
r = 1 mmE = ?
L = 1 m
dA || E so cos θ = 1
Compare to Finite Line Example From Last Week!Compare to Finite Line Example From Last Week!
If the Line Is Infinitely Long (L >> r) …Blah, Blah, Blah…
r
With Gauss’s Law We Got Same Answer With Two Lines of Algebra!
• Add the Vectors!
• Horrible Integral!
• Trig Substitution!
r
A-Rod
GaussGauss’’ Law: Insulating Plate Law: Insulating Plate
• Infinite INSULATING plane with uniform
charge density s
• E is NORMAL (perpendicular) to plane
• Construct Gaussian box as shown
For an insulator, E=σ/2ε0, and for a conductor, E=σ/ε0.
Surface Charge;σ = q/AUnits: [C/m2]
Surface Charge;σ = q/AUnits: [C/m2]
Recall Disk of Charged Sheet From Last Week!Recall Disk of Charged Sheet From Last Week!
• Add the Vectors!
• Horrible Integral!
• Trig Substitution!
• So Hard We Didn’t Do It!
If the Disk Has Large Radius (R>> z) …Blah, Blah, Blah…
With Gauss’s Law We Got Same Answer With Two Lines of Algebra!
Insulating and Conducting PlanesInsulating and Conducting Planes
Q
Insulating Plate: Charge Distributed Homogeneously.
Conducting Plate: Charge Distributed on the Outer Surfaces.Electric Field Inside a Conductor is ZERO!
Q/2
Two Insulating Sheets
The field from the plates cancels out so ignore them and use Coulombs inverse square law for the central charge only.
Two Conducting Sheets
4.867.68
12.54
E does not pass through a conductorFormula for E different by Factor of 2
7.68
4.86
GaussGauss’’ Law: Spherical Symmetry Law: Spherical Symmetry • Consider a POINT charge q & pretend
that you don’t know Coulomb’s Law• Use Gauss’ Law to compute the electric
field at a distance r from the charge• Use symmetry:
– place spherical surface of radius R centered around the charge q
– E has same magnitude anywhere on surface
– E normal to surface
r
q
Electric Fields With Spherical Electric Fields With Spherical Symmetry: Shell TheoremSymmetry: Shell Theorem
-15C
+10 CA spherical shell has a charge of +10C and a point charge of –15C at the center.What is the electric field produced OUTSIDE the shell?
If the shell is conducting?Field Inside a Conductor is ZERO!If the shell is conducting?Field Inside a Conductor is ZERO!
E
r
E=k(15C)/r2
E=k(5C)/r2E=0
And if the shell is insulating?
Charged Shells Behave Like a Point Charge of Total Charge “Q” at the CenterOnce Outside the Last Shell!
Charged Shells Behave Like a Point Charge of Total Charge “Q” at the CenterOnce Outside the Last Shell! Conducting
Electric Fields With Insulating SphereElectric Fields With Insulating Sphere
SummarySummary
• Electric Flux: a Surface Integral (Vector
Calculus!); Useful Visualization: Electric Flux
Lines Like Wind Through a Window.
• Gauss’ Law Provides a Very Direct Way to
Compute the Electric Flux.
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