Lecture 20

• calculating electric flux • electric flux through closed surface required for Gauss’s law: calculate E ¯ more easily; applies to moving charges • Uses of Gauss’s Law: charged sphere, wire, plane and conductor Calculating Electric Flux I

• Analogy: volume of air per second (m3/s) = vA= vA cos θ ¯ • Electric Flux (amount of E thru’ surface): Φe = EA= EAcos θ Area vector: A¯ = Anˆ • → Calculating Electric Flux II Φ = δΦ = E¯ . δA¯ e i e i i i → !Uniform E!¯, " # flat surface:

Φe = E.d¯ A¯ !surface = E cos θdA !... = E cos θ dA !... = E cos θA

Φe = E.d¯ A¯ !surface = EdA !... = E dA = EA !... Calculating Electric Flux III • Closed surface ( d A ¯ points toward outside: ambiguous for single surface): Φe = E.d¯ A¯ • strategy: divide !closed surface into either tangent or perpendicular to E¯

¯ 2 2 • example: cylindrical charge distribution, E = E0 r /r0 rˆ (ˆr in xy-plane) ! "

Φwall = EAwall

Φe = E.d¯ A¯ ! = Φtop + Φbottom + Φwall

= 0 + 0 + EAwall

= EAwall R2 = E (2πRL) 0 r2 " 0 # Flux due to point charge inside... • Gaussian surface with same symmetry as of charge distribution and hence E¯

Φe = E.d¯ A¯ = EAsphere (E¯ and same at all points) ⊥ ! q 2 E = 4π" r2 ; Asphere =4πr 0 q Φe = ⇒ "0 • flux independent of radius • approximate arbitrary shape... Φ = E.d¯ A¯ = q e !0 ! Charge outside..., multiple charges...Gauss’s Law

E¯ = E¯ + E¯ + ... (superposition) 1 2 ⇒ Φe = E¯1.dA¯ + E¯2.dA¯ + ... ! ! = Φ1 + Φ2 + ... q q = 1 + 2 ...for all charges inside ! ! " 0 0 # +(0 + 0 + ...for all charges outside) Q = q + q + ... for all charges inside in 1 2 ⇒ Using Gauss’s Law • Gauss’s law derived from Coulomb’s law, but states general property of E ¯ : charges create E ¯ ; net flux “flow”) thru’ any surface surrounding is same • quantitative: connect net flux to amount of charge Strategy • model charge distribution as one with symmetry (draw picture) symmetry of E¯ • Gaussian surface (imaginary) of same symmetry (does not have to enclose all charge) ¯ • E either tangent ( Φ e = 0) or perpendicular to ( Φ e = EA ) surface Charged Sphere: E¯ outside and inside • charge distribution inside has spherical symmetry (need not be uniform) Φ = E.d¯ A¯ = Qin e !0 EA = E4πr2 sphere! (don’t know E, but same at all points on surface) + Qin = Q E = Q 4π!0 (same⇒ as point charge) • flux integral not easy for other surface • using superposition requires 3D integral!

• spherical surface inside sphere Qin = Q ⇒ " Charged wire and plane • model as long line... Φe = Φtop + Φbottom + Φwall

= 0 + 0 + EAcyl. = E2πrL Φ = Qin ; Q = λL e !0 in λ Ewire = ⇒ 2π!0r • independent of L of imaginary... • cylinder encloses only part of wire’s charge: outside does not contribute to flux, but essential to cylindrical symmetry (easy flux integral); cannot use for finite length (E ¯ not same on wall) • Gauss’s law effective for highly symmetrical: superposition always works... Conductors in Electrostatic Equilibrium: E ¯ at surface ¯ •Ein =0 if not, charges (free to move ) would... net charge E ¯ =0 outside • → " • If E ¯ tangent to surface, charges move... Φe = AEsurface for outside face +0 for inside face (E¯in = 0) +0 for wall (E¯ surface) Qin ⊥ Φe = ; Qin = ηA !0 ⇒ η E¯surface = , to surface !0 ⊥ ! " Within conductor...

• excess charge on exterior surface • E ¯ =0 inside hole ( E¯ =0 inside conductor and no charge in hole): screening • charge inside hole of neutral conductor polarizes...