Gausss Law The first Maxwell Equation A very useful computational - - PowerPoint PPT Presentation

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Gausss Law The first Maxwell Equation A very useful computational - - PowerPoint PPT Presentation

Oct 25, 2022 281 likes •434 views

Gausss Law The first Maxwell Equation A very useful computational technique This is important! P05 - 7 Gausss Law The Idea The total flux of field lines penetrating any of these surfaces is the same and depends only on the

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SLIDE 1

7 P05 -

Gauss’s Law

The first Maxwell Equation A very useful computational technique This is important!

slide-2
SLIDE 2

8 P05 -

Gauss’s Law – The Idea

The total “flux” of field lines penetrating any of these surfaces is the same and depends only

  • n the amount of charge inside
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SLIDE 3

9 P05 -

Gauss’s Law – The Equation

S surface closed

ε

in E

q d = ⋅ = Φ

∫∫

A E

  • Electric flux ΦE (the surface integral of E over

closed surface S) is proportional to charge inside the volume enclosed by S

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SLIDE 4

10 P05 -

Now the Details

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SLIDE 5

11 P05 -

Electric Flux ΦE

Case I: E is constant vector field perpendicular to planar surface S of area A

∫∫

⋅ = Φ A E

  • d

E

E

EA Φ = +

Our Goal: Always reduce problem to this

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SLIDE 6

12 P05 -

Electric Flux ΦE

Case II: E is constant vector field directed at angle θ to planar surface S of area A

cos

E

EA θ Φ =

∫∫

⋅ = Φ A E

  • d

E

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SLIDE 7

14 P05 -

Gauss’s Law

S surface closed

ε

in E

q d = ⋅ = Φ

∫∫

A E

  • Note: Integral must be over closed surface
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SLIDE 8

15 P05 -

Open and Closed Surfaces

A rectangle is an open surface — it does NOT contain a volume A sphere is a closed surface — it DOES contain a volume

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SLIDE 9

16 P05 -

Area Element dA: Closed Surface

For closed surface, dA is normal to surface and points outward ( from inside to outside)

ΦE > 0 if E points out ΦE < 0 if E points in

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SLIDE 10

21 P05 -

Electric Flux: Sphere

Point charge Q at center of sphere, radius r E field at surface:

2

ˆ 4 Q r πε = E r

  • Electric flux through sphere:

2 S

ˆ ˆ 4 Q dA r πε = ⋅

∫∫

r r

  • S

E

d Φ = ⋅

∫∫ E

A

  • r

A ˆ dA d =

  • Q

ε =

2 2

4 4 Q r r π πε =

2 S

4 Q dA r πε =

∫∫

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SLIDE 11

22 P05 -

Arbitrary Gaussian Surfaces

S surface closed

ε Q d

E

= ⋅ = Φ

∫∫

A E

  • For all surfaces such as S1, S2 or S3
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SLIDE 12

36 P05 -

Gauss: Planar Symmetry

Infinite slab with uniform charge density σ Find E outside the plane

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SLIDE 13

37 P05 -

Gauss: Planar Symmetry

Symmetry is Planar Use Gaussian Pillbox

x E ˆ E ± =

  • x

ˆ

Gaussian Pillbox

Note: A is arbitrary (its size and shape) and should divide out

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SLIDE 14

38 P05 -

Gauss: Planar Symmetry

A qin σ =

Total charge enclosed: NOTE: No flux through side of cylinder, only endcaps

( )

2

in

q A E A σ ε ε = = =

S S E Endcaps

d E dA EA Φ = ⋅ = =

∫∫ ∫∫

E A

  • +

+ + + + + + + + + + +

σ E

  • E
  • x

A

{ }

ˆ to right ˆ

  • to left

2 σ ε ⇒ = x E x

  • 2

E σ ε =