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.