Chapter 23: Gauss’ Law
PHY2049: Chapter 23
1
Two Equivalent Laws of Static Electricity
Equivalent!
Coulomb’s Law
Gauss’ Law
Derivation given in Sec. 23-5
(Read yourself)
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 Nm2/C
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
2
ˆ
ˆ
A = ( 2i + 3 j ) m
2
ˆ
ˆ
ˆ
Φ E ≡ E ⋅ A = 4i ⋅ ( 2i + 3 j ) = 8 Nm / C
PHY2049: Chapter 23
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Flux Through Closed Surface
ÎSurface
elements
dA always point
outward!
ÎAs
a result, sign
of ΦE is
‹E outward (+)
‹E inward (−)
ΦE < 0
ΦE > 0
ΦE = 0
PHY2049: Chapter 23
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Example: Flux Through Cubic Surface
ÎE
r
E = Ezˆ
field is constant:
‹Flux through front
face?
‹Flux through back
face?
‹Flux through top
face?
‹Flux through whole
cube?
PHY2049: Chapter 23
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Example: Flux Through Cylindrical Surface
ÎAssume
E is constant, to the right
‹Flux through left face?
‹Flux through right face?
‹Flux through curved side
‹Total flux through cylinder?
PHY2049: Chapter 23
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Example: Flux Through Spherical Surface
ÎPoint
ÎE
charge +Q at center
points radially outward (normal to surface!)
Φ E = ∫ E ⋅ dA
S
(
 kQ 
=  2  4π r 2
r 
Q
= 4π kQ =
)
ε0
Does not depend on the
radius of the sphere!
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
ÎCan
see)
be useful in finding charge distribution (as we shall
ÎConsequences
of Gauss’ law (as we shall see)
‹ Excess
charge on conductor is always on surface
Conductor
‹ E is always normal to surface on 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) 0
‹2) +Q/ε0
‹3)
- Q/ε0
‹4)
+Q
‹5)
-Q
PHY2049: Chapter 23
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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
11
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 ones have variable E fields?
ÎWhat are the fluxes through 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
‹
+Q at a point
By symmetry, E must be radially
symmetric
ÎDraw
point
Gaussian surface around
Sphere of radius r
‹ E field has constant magnitude,
⊥ to
Gaussian surface
‹
Q
∫SE ⋅ dA = E (4πr ) = ε0
2
Gauss’ Law
Solve for E
E=
Q
4πε 0 r
2
=
kQ
r2
PHY2049: Chapter 23
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Example
ÎCharges
on a ball and spherical shells, each
uniformly charged:
‹+Q (ball at center)
‹+3Q (middle shell)
‹- 2Q (outer shell)
ÎFind
fluxes through the
three Gaussian surfaces
(a) Inner +Q/ε0
(b) Middle +4Q/ε0
(c) Outer +2Q/ε0
PHY2049: Chapter 23
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