Chapter 23: Gauss’ Law
PHY2049: Chapter 23
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Two Equivalent Laws for Electricity
equivalent
Coulomb’s Law
Gauss’ Law
Derivation given in Sec. 23-5 (Read!)
Not derived in this book (Requires vector
calculus)
We will focus on what is Gauss’ law and how we use it.
PHY2049: Chapter 23
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Electric Flux
ÎSimple
‹E
definition of electric flux (E constant, flat surface)
at an angle θ to planar surface, area A
Φ E ≡ E ⋅ A = EA cos θ
‹ Units
ÎSimple
= N m2 / C (SI units)
Normal
E
example
‹ Let
E = 104 N/C pass through 2m x 5m rectangle, 30° to normal
‹ φE = 104 * 10 * cos(30°) = 100,000 * 0.866 = 86,600
ÎMore
general ΦE definition (E variable, curved surface)
Φ E ≡ ∫ E ⋅ dA
S
PHY2049: Chapter 23
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Example of Constant Field
E = 4iˆ N/C
A = (2iˆ + 3 ˆj ) m 2
Φ E ≡ E ⋅ A = 4iˆ ⋅ ( 2iˆ + 3 ˆj ) = 8 Nm 2 / C
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Flux Through Closed Surface
ÎSurface
elements dA
always point outward!
ÎSign
of ΦE
‹E
outward (+)
‹ E inward (−)
ΦE < 0
ΦE > 0
ΦE = 0
PHY2049: Chapter 23
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Example: Flux Through Cube
r
ÎE field is constant: E = Ezˆ
‹ Flux
through
‹ Flux through
‹ Flux through
‹ Flux through
front face?
back face?
top face?
whole cube?
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Example: Flux Through Cylinder
ÎAssume
E is constant, to the right
‹ Flux
through left face?
‹ Flux through right face?
‹ Flux through curved side
‹ Total flux through cylinder?
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Example: Flux Through Sphere
ÎAssume
ÎE
point charge +Q
points radially outward (normal to surface!)
Φ E = ∫ E ⋅ dA
S
(
 kQ 
=  2  4π r 2
r 
Q
= 4π kQ =
)
ε0
Foreshadowing of Gauss’ Law!
PHY2049: Chapter 23
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Gauss’ Law
ÎGeneral
statement of Gauss’ law
qenc
∫SE ⋅ dA = ε0
ÎCan
Integration over closed surface
qenc is charge inside the surface
Charges outside surface have no effect
(This does not mean they do not
contribute to E.)
be used to calculate E fields. But remember
‹ Outward
E field, flux > 0
‹ Inward E field, flux < 0
ÎConsequences
of Gauss’ law (as we shall see)
‹ Excess
charge on conductor is always on surface
‹ E is always normal to surface on conductor
Conductor
(Excess charge distributes on surface in such a way)
PHY2049: Chapter 23
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Reading Quiz
ÎWhat
is the electric flux through a sphere of radius R
surrounding a charge +Q at the center?
‹ 1)
‹ 2)
‹ 3)
‹ 4)
‹ 5)
0
+Q/ε0
−Q/ε0
+Q
None of these
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Question
dS
dS
+Q
+Q
2
1
How does the flux ФE through the entire surface
change when the charge +Q is moved from position
1 to position 2?
a) ФE increases
ФE decreases
c) ФE doesn’t change
b)
PHY2049: Chapter 23
Just depends on charge
not position
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Power of Gauss’ Law: Calculating E Fields
ÎValuable
‹E
for cases with high symmetry
= constant, ⊥ surface
∫ E ⋅ dA = ± EA
S
‹E
|| surface
ÎSpherical
symmetry
∫ E ⋅ dA = 0
S
‹E
field vs r for point charge
‹ E field vs r inside uniformly charged sphere
‹ Charges on concentric spherical conducting shells
ÎCylindrical
symmetry
‹E
field vs r for line charge
‹ E field vs r inside uniformly charged cylinder
ÎRectangular
symmetry
‹E
field for charged plane
‹ E field between conductors, e.g. capacitors
PHY2049: Chapter 23
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Example
Î4
Gaussian surfaces: 2 cubes and 2 spheres
ÎRank
magnitudes of E field on surfaces
ÎWhich
ÎWhat
ones have variable E fields?
are the fluxes over each of the Gaussian surfaces
(a) E falls as radius increases
(b) E non-constant on cube (r changes)
(c) Fluxes are same, +Q/ε0
PHY2049: Chapter 23
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Derive Coulomb’s Law From Gauss’ Law
ÎCharge
‹ By
ÎDraw
+Q at a point
symmetry, E must be radially symmetric
Gaussian’ surface around point
r
‹ Sphere
of radius r
‹ E field has constant mag., ⊥ to Gaussian surface
Q
∫SE ⋅ dA = E (4πr ) = ε 0
2
E=
Q
4πε 0 r
2
=
kQ
r
2
Gaussian surface
(sphere)
Gauss’ Law
Solve for E
PHY2049: Chapter 23
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Example
ÎCharges
on shells are
‹ +Q
(ball at center)
‹ +3Q (middle shell)
‹ +5Q (outside shell)
ÎFind
fluxes on the three Gaussian surfaces
(a) Inner +Q/ε0
(b) Middle +4Q/ε0
(c) Outer +9Q/ε0
PHY2049: Chapter 23
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Conductors with No Current
ÎE
is zero everywhere inside
Why? Conductors are full of mobile charges (e.g., conduction
electrons in a background formed by immobile positive ions). If there
were E, then the charges must be moving around due to force F=qE.
This would contradict “no current.”
Note: even if there is an externally imposed E, it cannot go inside
ÎAll
excess charge must be on outer surface.
Why? Since E=0 everywhere inside, qenc enclosed by any Gaussian
surface is also zero everywhere inside.
Note: distribution of surface charge must be such to make E=0
everywhere inside
ÎE
is always normal to surface on conductor
Why? E component parallel to surface would cause surface charge to
move. This would contradict “no current.”
PHY2049: Chapter 23
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