Lecture 20
• calculating electric flux
• electric flux through closed surface
required for Gauss’s law: calculate Ē 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) = v A = vA cos θ
Electric Flux (amount of Ē thru’ surface): Φe = E A = EA cos θ
Area vector: Ā = An̂ →
Calculating Electric Flux II
Φe =
!
i
δΦe =
!
i
Ēi . δ Ā
Uniform Ē,
flat surface:
Φe
=
!
Ē.dĀ
surf ace
=
!
E cos θdA
...
!
= E cos θ
dA
...
= E cos θA
Φe
=
!
Ē.dĀ
surf ace
=
!
EdA
...
!
= E
dA = EA
...
"
#
i
→
•
•
•
Calculating Electric Flux III
Closed surface !( dĀ points toward outside: ambiguous for single
surface): Φe = Ē.dĀ
strategy: divide closed surface into either tangent or
perpendicular to Ē
example: cylindrical charge distribution,
Ē = E0 r
Φwall = EAwall
Φe
=
!
Ē.dĀ
= Φtop + Φbottom + Φwall
=
0 + 0 + EAwall
= EAwall
#
"
2
R
=
E0 2 (2πRL)
r0
!
2
/r02
"
r̂ (r̂ in xy-plane)
Flux due to point
charge inside...
•
Gaussian surface with
same symmetry as of
charge distribution
and hence Ē
!
Φe = Ē.dĀ = EAsphere
(Ē ⊥ and same at all points)
E = 4π"q0 r2 ; Asphere = 4πr2
q
⇒ Φ e = "0
•
flux independent of
radius
•
approximate arbitrary
shape...
!
q
Φe = Ē.dĀ = !0
Charge outside..., multiple charges...Gauss’s Law
Ē = Ē1 +!Ē2 + ... (superposition)
⇒
!
Φe =
Ē1 .dĀ + Ē2 .dĀ + ...
Qin
= Φ1 + Φ2 + ...
"
#
q1
q2
=
+ ...for all charges inside
!0
!0
+(0 + 0 + ...for all charges outside)
= q1 + q2 + ... for all charges inside ⇒
Using Gauss’s Law
•
Gauss’s law derived from Coulomb’s law, but states general
property of Ē : charges create Ē ; 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 Ē
•
Gaussian surface (imaginary) of same symmetry (does not have
to enclose all charge)
•
Ē either tangent (Φe = 0) or perpendicular to (Φe = EA) surface
•
Charged Sphere: Ē outside and inside
charge distribution inside has
spherical symmetry (need
not be! uniform)
Q
Φe = Ē.dĀ = !in
0
EAsphere = E4πr2
(don’t know E, but
same at all points on surface)
+ Qin = Q
Q
⇒ E = 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
• Φmodel= asΦlong+ line...
Φ
e
top
=
bottom
+ Φwall
0 + 0 + EAcyl.
= E2πrL
Qin
!0 ;
Φe =
Qin = λL
⇒ Ewire = 2π!λ0 r
•
•
independent of L of imaginary...
•
Gauss’s law effective for highly
symmetrical: superposition
always works...
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 (Ē not same on wall)
Conductors in Electrostatic Equilibrium: Ē at surface
•Ē = 0if not, charges (free to move ) would...
• net charge → Ē "= 0 outside
to surface, charges move...
• If ĒΦ tangent
= AE
for outside face
in
e
surf ace
+0 for inside face (Ēin = 0)
+0 for wall (Ē ⊥ surface)
; Qin = ηA ⇒
Φe = Q!in
0
!
"
Ēsurf ace = !η0 , ⊥ to surface
Within conductor...
•
•
•
excess charge on exterior surface
Ē = 0 inside hole (Ē = 0 inside
conductor and no charge in hole):
screening
charge inside hole of neutral
conductor polarizes...