C2 2 Gauss Law

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!) 電磁9701-c2.2-2

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:

Electric flux through closed surface S =

(electric charge enclosed by surface S)

ε0

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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: 電磁9701-c2.2-10

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 電磁9701-c2.2-11

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’ Law a) Charged sphere – concentric Gaussian sphere、spherical coordinates b) Charged cylinder – coaxial Gaussian cylinder、cylindrical coordinates c) Charged box / Charged plane – use rectangular box、rectangular coordinates d) 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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APPLICATIONS OF GAUSS’ LAW - Example 2.2 -2/3 • Gauss’ law

(by symmetry of sphere) (for Gaussian sphere)

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

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

or

The electric field (for r > R) for charged sphere is equivalent to that of a point charge q located at the origin!!! 電磁9701-c2.2-15

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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Example 2.3 (conti.) • from cylindrical symmetry = constant on cylindrical Gaussian surface • What are the vector area element?

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Example 2.3 (conti.)

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Example 2.3 (conti.) 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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Example 2.3 (conti.) • Putting this all together now: where

=>

or

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Example 2.3 (conti.) 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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Example 2.3 (conti.) • Enclosed charge (for s > S): • symmetry of long cylinder

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Example 2.3 (conti.) • vector area element

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Example 2.3 (conti.) • Now • Then

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

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Example 2.3 (conti.) • 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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Example 2.4 (conti.) • from the symmetry associated with ∞-plane

• six sides and six outward unit normal vectors:

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Example 2.4 (conti.) • Then

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Example 2.4 (conti.) • since

• The integrals is separated into two regions:

• Then

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Example 2.4 (conti.)

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

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Example 2.4 (conti.) • Thus, we have:

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

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Example 2.4 (conti.) • Now, what is Qencl?

• Note: 電磁9701-c2.2-33

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): 電磁9701-c2.2-37

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 DISTRIBUTION STATIC = NO TIME DEPENDENCE / VARIATION 電磁9701-c2.2-39

Helmholtz theorem • Vector field A(r) is fully specified if both its divergence and its curl are known. • Corollary: Any differentiable vector function A(r) that goes to zero faster than 1 /r as r → ∞ can be expressed as 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 ( Electrostatic Potential )

~ valid for localized charge distributions ~ NOT vaild for infinite-expanse charge distributions 電磁9701-c2.2-41