c2 2 Gauss Law

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Gauss’ Law – electric flux

• Consider an imaginary sphere of radius R centered on charge Q at origin:

• FLUX OF ELECTRIC FIELD LINES ?

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Gauss’ Law – total electric flux

• FLUX OF ELECTRIC FIELD LINES (through surface S):

• ΦE = “measure” of “number of E-field “lines”passing through surface S, (SI Units: Volt-meters).

• TOTAL ELECTRIC FLUX (ΦETOT )

associated with any closed surface S, is a measure of the (total) charge enclosed by surface S.

• Charge outside of surface S will contribute nothing to total electric flux ΦE (since E-field lines pass through one portion of the surface S and out another – no net flux!)

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Gauss’ Law - calculation

• Consider point charge Q at origin. • Calculate ΦE passing through a sphere of radius r:

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Gauss’ Law in integral form

• Gauss’ Law in integral form:

0

S) surfaceby enclosed charge (electric

S surface closedgh flux throu Electric

ε=

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Gauss’ Law – discrete charges

• If ∃ (= there exists) lots of discrete charges qi , all enclosed by Gaussian surface S’

• by principle of superposition

• Then

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Gauss’ Law – volume charge

• If ∃ volume charge density ρ(r) , then

• Using the DIVERGENCE THEOREM:

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Gauss’ Law in Differential Form

• This relation holds for any volume v ⇒ the integrands of ∫v ( ) dτ' must be equal

• So, Gauss’ Law in Differential Form:

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The DIVERGENCE OF E(r) – 1/3

• Calculate from Coulomb’ law

Extend over all space !!

NOT a constant !!

• r : Field point P• r’: source point S

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The DIVERGENCE OF E(r) – 2/3

• Recall that

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The DIVERGENCE OF E(r) -3/3

Thus

or

Now, we have the Gauss’s law:

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Gauss’ Law in Integral Form - revisited

• By the differential form of Gauss’ law:

• We have,

• Apply the Divergence Theorem

• Thus: Gauss’ Law in Integral Form

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GAUSS’ LAW AND SYMMETRY

• Use of symmetry can be extremely powerful in terms of simplifying seemingly complicated problems.

• Examples of use of Geometrical Symmetries and Gauss’ Lawa) Charged sphere

– concentric Gaussian sphere、spherical coordinatesb) Charged cylinder

– coaxial Gaussian cylinder、cylindrical coordinatesc) Charged box / Charged plane

– use rectangular box、rectangular coordinatesd) Charged ellipse

– concentric Gaussian ellipse、 elliptical coordinates

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APPLICATIONS OF GAUSS’ LAW - Example 2.2 -1/3

• Find / determine the electric field intensity E(r) outside a uniformly charged solid sphere of radius R and total charge q

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• Gauss’ law

(by symmetry of sphere)

(for Gaussian sphere)

• By symmetry, the magnitude of E is constant ∀ any fixed r !!

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or

The electric field (for r > R) for charged sphere is equivalent to that of a point charge q located at the origin!!!

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Example 2.3

• Consider a long cylinder of length L and radius S that carries a volume charge density ρ that is proportional to the distance from the axis s of the cylinder, i.e.

a) Determine the electric field E(r) inside this long cylinder.

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Example 2.3 (conti.)

• Gauss’ law :

• Enclosed charge :

• By cylindrical Symmetry

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• from cylindrical symmetry

= constant on cylindrical Gaussian surface

• What are the vector area element?

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Note:• On LHS and RHS endcaps E(r) is not constant, because

r is changing there• However, note that

=> Gaussian endcap terms do not contribute!!!

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• Putting this all together now:

where

or

=>

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b) Find ELECTRIC FIELD E(r) outside of this long cylinder

• use Coaxial Gaussian cylinder of length l (<< L) and radius s (> S):

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• Enclosed charge (for s > S):

• symmetry of long cylinder

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• vector area element

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• Now• Then

∴ Electric field outside charged rod (s = r > S) :

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• Inside (s < S): • Outside (s > S):

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Example 2.4

• An infinite plane carries uniform charge σ. Find the electric field.

• Use Gaussian Pillbox centered on ∞-plane:

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• from the symmetry associated with ∞-plane

• six sides and six outward unit normal vectors:

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• Then

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• since

• The integrals is separated into two regions:

• Then

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So, non-zero contributions are from bottom and top surfaces

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• Thus, we have:

• These integrals are not over z, and E(z) = constant for z = zowe can pull E(z) outside integral,

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• Now, what is Qencl ?

• Note:

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Curl of E(r)

• Consider point charge at origin:

• By spherical symmetry (rotational invariance)1. E(r) is radial.2. thus static E-field has no curl.

• Calculation : see next page

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Curl of E(r)

• Let’s calculate:

• In spherical coordinates:

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Curl of E(r)

• thus

• So, around a closed contour C

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Curl of E(r)

• Use Stokes’ Theorem

• Since must be true

for arbitrary closed surface S,

this can only be true for all ∀ closed surfaces S IFF (if and only if):

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Curl of E(r) - discrete charges

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Curl of E(r) - discrete charges

• It can be shown that

FOR ANY STATIC CHARGE DISTRIBUTIONSTATIC = NO TIME DEPENDENCE / VARIATION

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Helmholtz theorem

• Vector field A(r) is fully specified if both its divergenceand its curl are known.

• Corollary:Any differentiable vector function A(r) that goes to zero faster than 1 /r as r → ∞ can be expressedas the gradient of a scalar plus the curl of a vector:

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Helmholtz theorem for electrostatics

• For the case of electrostatics:

• Thus

• i.e.

with

~ valid for localized charge distributions~ NOT vaild for infinite-expanse charge distributions

( Electrostatic Potential )

c2 2 Gauss Law

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