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Dr. Hugh Blanton ENTC 3331
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Dr. Hugh Blanton
ENTC 3331
Gauss’s Law
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 3
• Recall• Divergence literally means to get farther
apart from a line of path, or• To turn or branch away from.
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 4
• Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles:
Goes straight ahead at constant velocity.
(degree of) divergence 0
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 5
Now suppose they turn with a constant velocity
diverges from original direction
(degree of) divergence 0
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 6
Now suppose they turn and speed up.
diverges from original direction
(degree of) divergence >> 0
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 7
Current of water
No divergence from original direction
(degree of) divergence = 0
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 8
Current of water
Divergence from original direction
(degree of) divergence ≠ 0
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 9
• Source• Place where something originates.• Divergence > 0.
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 10
• Sink• Place where something disappears.• Divergence < 0.
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 11
• Derivation of Divergence Theorem• Suppose we have a cube that is infinitesimally small.
one of six faces
in
Vector field, V(x,y,z)
x
y
z
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 12
• Need the concept of flux:• water through an area• current through an area
• water flux per cross-sectional area (flux density implies• (total) flux = = scaler.
j
A
A
Aj ˆˆ
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 13
• Let’s assume the vector, V(x,y,z), represents something that flows, then• flux through one face of the cube is:
• For example might be:
• and
inV ˆ
in
xn ˆˆ dydzyz
dydzVdydzV xx xx ˆˆ
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 14
• The following six contributions for each side of the cube are obtained:
dydzVx
dydzVxdxdzVy
dxdzVy
dxdyVz
dxdyVz
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 15
• Now consider the opposite faces of the infinitesimally small cube.
• This holds equivalently for the two other pairs of faces.
in
x
y
z
dxx
VVV x
xx
112
dx
2xV1xV differential change of Vx over dx
vector magnitude on the input side.
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 16
in
x
y
z
dxx
VVV x
xx
12
dx
2xV1xV
dydzVdydzV xx xx ˆˆand
dydzdxx
VVdydzdx
x
VV x
xx
x
21
1
• Flux in the x-direction.
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 17
Divergence TheoremDivergence Theorem
• Divergence Theorem• Gauss’s Theorem• Valid for any vector field• Valid for any volume,
• Whatever the shape.
zyx Vz
Vy
Vx
VVdiv
Note that the above only applies to the Cartesian coordinate system.
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 18
• Since Gauss’s law can be applied to any vector field, it certainly holds for the electric field, and the electric flux density, .
• The use of in this context instead of is historical.
zyx ,,E
zyx ,,D
sdDdVDSV
ˆ
D
E
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 19
• If Gauss’s law is true in general, it should be applicable to a point charge.• Constuct a virtual sphere around a positive
charge with radius, R.
• must be radially outward along the unit vector, .
+q
D
D
R
sd
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 20
SSS
sDdsdDd RRsD ˆˆˆ
SSS
ddRDsdDsDd sin2
0
2
0
2 sin ddDR
0
2 sin2 dDR
220
2 4112cos2 DRDRDR
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 21
• What about the volume integral?
• only has a component along the radius vector
D
RD
RDR
RRdV R
V sin
1sin
sin
11 22
D
D
RD
RDR
RR R sin
1sin
sin
11 22
D
0,0,0 DDDR
D
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 22
RDRRR
22
1
D
2
00 0
222
sin1
dddRRDRRR
dVDR
R
V
4
R
R
R
R dRDRR
dRRDRRR 0
2
0
222
41
4
What is this?
44
1 22
qRD
R
qED R
ooRoR
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 23
• Throw in some physics!
24
1
R
qED RR
qqdRR
qdRR
qR
R
RR R
00 022 14
4
integration and differentiation cancel out
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 24
• So what?
• Coulomb’s law and Gauss’s law are equivalent for a point charge!
qdVdDRVS
DsD
ˆ4 2
qRo 24 E
24
1
R
q
oE
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 25
qdVdDRVS
DsD
ˆ4 2
divergence theorem
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 26
qdVdDRVS
DsD
ˆ4 2
Gauss’s Law
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 27
• Because of its greater mathematical versatility, Gauss’s law rather than Coulomb’s law is a fundamental postulate of electrostatics.• A postulate is believed to be true, although no
proof may be possible.
• Any surface of an arbitrary volume.
QdVdVS
DsD
ˆ
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 28
• Note
• which infers
sDD ˆddVQdVSVV
V
definition of charge distribution
Gauss’s Law
V D Differential form of
Gauss’s Law
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 29
• Maxwell Equation
• One of two Maxwell equations for electrostatics.
V D
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 30
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 31
t
DJH
t
BE
B
D
0
Electric flux density orDisplacement Field [C/m2]
Charge Density [C]
Magnetic Induction [Weber/m2
or Tesla]]
Magnetic Field [A/m] Current Density [A/m2]
Electric Field [V/m]Time [s]
Page 139
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 32
t
EJH
t
HE
H
E
r
r
r
r
)(
)(
0)(
)(
0
0
0
0
Page 139
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 33
• Use Gauss’s law to obtain an expression for the E-field from an infinitely long line of charge.
constantl 0
X)(rE
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 34
• Symmetry Conditions• Infinite line of charge• • •
0 ED0 zz ED
rrr DD
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 35
• Gauss’s law considers a hypothetical closed surface enclosing the charge distribution.• This Gaussian surface can have any shape,
but the shape that minimizes our calculations is the shape often used.
constantl 0
D sdQd
S
sD ˆ
h
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 36
• The total charge inside the Gaussian volume is:
• The integral is:
• The right and left surfaces do not contribute since.
hQ l
dzrdDdh
o r
S
rrsD ˆˆˆ2
0
0zD
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 37
2
0
h
or dzdrD
rhDr2
and
hrhD lr 2
rol
r Er
D
2
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 38
• Two infinite lines of charge.• Each carrying a charge density, l.• Each parallel to the z-axis at
• x = 1 and x = -1.
• What is the E-field at any point along the y-axis?
constantl
constantl
x
z
1
1
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 39
• For a single line of constant charge
• Using the principle of superposition of fields:
rE
o
lr
2
21 EEE
tot
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 40
x
x
y
z
1-1
1r2r
)0,,0( yE
2221 1010 yyr
11
ˆˆˆ
r
yxr
y
2222 1010 yyr
22
ˆˆˆ
r
yxr
y
22 1
ˆˆ
1
ˆˆ
2 y
y
y
y
o
ltot
yxyxE
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 41
• Only interested in the y-component of the field
22 1
ˆ
1
ˆ
2 y
y
y
y
o
ltot
yyE
21
ˆ2
2 y
y
o
ltot
yE
21
ˆ
y
y
o
ltot
yE
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 42
• A spherical volume of radius a contains a uniform charge density V.
• Determine for • and•
ED
aR aR
+q
DsdNote: Charge distribution for
an atomic nucleus where a = 1.210-15 m A⅓ (A is the mass number)
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 43
• Outside the sphere (R a), use Gauss’s Law
• To take advantage of symmetry, use the spherical coordinates:
• and
S
dsD ˆ
ddRds sin2
rD ˆrD
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 44
• Field is always perpendicular for any sphere around the volume.
• The left hand side of Gauss’s Law is
0
2
0
22 sinsinˆˆˆ ddRDddRDd R
S
R
S
RRsD
22
44
R
QDQRD RR
4
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 45
• Recall that
dVdVV
V
V D
V D
0
2
0 0
2 sina
V
V
V ddRdRdV
0
2
0 0
2sina
V
V
V dRRdddV
4
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 46
Qa
dRR V
a
V 344
3
0
2
2
3
2
3
2 334
4
4 R
a
R
a
R
QD VV
R
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 47
• Inside the sphere (R a), use Gauss’s Law
QdVdV
V
S
sD ˆ
24 RDRpreviously calculated
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 48
R
V
V
V ddRdRdV0 0
2
0
2 sin
R
V dRR0
24
3
4 3RV
3
44
32 R
RD VR
3
RD V
R
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 49
• Thin spherical shell• Find E-field for• and •
aR aR
constantS
0S
a
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 50
• Inside ( )• Gauss’s Law
• This is only possible if .
aR
constantS
0S
a0ˆ Qd
S
sD
0D
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 51
• Outside ( )• Gauss’s Law
aR
constantS
0S
a S
S
S
dSQd sD ˆ
0
2
0
2 sin ddadS S
S
S
24 adS S
S
S
24 RDRpreviously calculated
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 52
22 44 aRD SR
2
2
R
aD SR
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 53
• An electric field is given as
• Determine• • Q in a 2m 2m 2m cube.
mVyxyxzyx 23ˆ2ˆ
1,, yxE
V
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 54
• Maxwell’s equation of Electrostatics
z
x
y
ED
divdiv V
yxy
yxx
div 232
E
0022 Vdiv E
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 55
z
x
y
QdS
sD ˆ
dydzyx xx ˆˆ22
0
2
0
For the surface 1 directed in the x-direction.
dydzyx 2
0
2
02
2
0
22
0
2
0
20 2
442
yxydyyxdyzyx
1
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 56
882
42
0
2
x
yxy
z
x
y
1
2
For the surface 2 directed in the -x-direction.
dydzyx xx ˆˆ22
0
2
0
8822
0
2
0 xdydzyx
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 57
z
x
y
3
For the surface 3 & 4 directed in the z- & -z directions.
40ˆ
S
dsD
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 58
z
x
yFor the surface 5 directed in the y-direction.
5
dxdzyx yy ˆˆ232
0
2
0
dxdzyx 2
0
2
023
dxyxdxyzxz 2
0
2
0
20 4623
yxyx
81242
62
0
2
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 59
z
x
yFor the surface 6 directed in the -y-direction.
6
dxdzyx yy ˆˆ232
0
2
0
dxdzyx 2
0
2
023
y812
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 60
• By superposition
• Indeed, there is no charge in the cube.
0ˆ S
dsD
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 61
• Find in all regions of an infinitely long cylindrical shell.• Inner shell( )• Cylindrical volume.
D
0V
constantV
3
1
1r
0ˆ QdS
sD
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 62
• Shell itself ( )• Cylindrical coordinates.
0V
constantV
3
1
31 r
dzrdDdh
r
S
rrsD ˆˆˆ2
0 0
r
D sd
h
r
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 63
dzrdDdzrdDh
r
h
r
2
0 0
2
0 0ˆˆ rr
rhDdrhDdzdrD rr
h
r
22
0
2
0 0
• Top and bottom face of cylinder do not contribute to .D
r h
V dzrdrd1
2
0 0
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 64
drrhdzrdrdr
V
r h
V 11
2
0 02
12
2 2
1
2
rh
rh V
r
V
V
V
S
dVd sD ˆ
)1(2 2 rhrhD Vr
Dr. Blanton - ENTC 3331 - Gauss’s Theorem 65
)1(2 2 rhrhD Vr
r
rD V
r
)1(
2
2
