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

Gauss’s Law The first Maxwell Equation A very useful computational technique This is important! P05 - 7

Gauss’s Law – The Idea The total “flux” of field lines penetrating any of these surfaces is the same and depends only on the amount of charge inside P05 - 8

Gauss’s Law – The Equation � � q ∫∫ Φ = ⋅ = in E A d ε E 0 closed surface S Electric flux Φ E (the surface integral of E over closed surface S) is proportional to charge inside the volume enclosed by S P05 - 9

Now the Details P05 - 10

Electric Flux Φ E Case I: E is constant vector field perpendicular to planar surface S of area A � � ∫∫ Φ = ⋅ E d A E Φ = + EA E Our Goal: Always reduce problem to this P05 - 11

Electric Flux Φ E Case II: E is constant vector field directed at angle θ to planar surface S of area A � � ∫∫ Φ = ⋅ E A d E Φ = θ EA cos E P05 - 12

Gauss’s Law � � q ∫∫ Φ = ⋅ = in E A d ε E 0 closed surface S Note : Integral must be over closed surface P05 - 14

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 P05 - 15

Area Element dA: Closed Surface For closed surface, d A is normal to surface and points outward ( from inside to outside) Φ E > 0 if E points out Φ E < 0 if E points in P05 - 16

Electric Flux: Sphere Point charge Q at center of sphere, radius r E field at surface: � Q = ˆ E r πε 2 4 r 0 Electric flux through sphere: � � Q ∫∫ E ∫∫ Φ = ⋅ = ⋅ � � ˆ ˆ d A r dA r πε E 2 4 r 0 S S � = Q Q Q ˆ ∫∫ d A dA r = � = π = 2 dA 4 r πε πε ε 2 2 4 r 4 r 0 0 0 S P05 - 21

Arbitrary Gaussian Surfaces � � Q ∫∫ Φ = ⋅ = E d A ε E 0 closed surface S For all surfaces such as S 1 , S 2 or S 3 P05 - 22

Gauss: Planar Symmetry Infinite slab with uniform charge density σ Find E outside the plane P05 - 36

Gauss: Planar Symmetry Symmetry is Planar � = ± ˆ E x E Use Gaussian Pillbox ˆ x Note: A is arbitrary (its size and shape) and Gaussian should divide out Pillbox P05 - 37

Gauss: Planar Symmetry = σ q in A Total charge enclosed: NOTE: No flux through side of cylinder, only endcaps � � ∫∫ ∫∫ Φ = ⋅ = = � � + E d A E dA EA x E Endcaps + S S + σ + q A ( ) = = = in E 2 A + ε ε � � + A 0 0 + E E { } + � σ σ ˆ x to right + = ⇒ = E E + ε ε ˆ - x to left 2 2 + 0 0 + σ P05 - 38