del operator
DESCRIPTION
all abt del operator in electromagneticsTRANSCRIPT
Dr. Kent Chamberlin 1
The Del OperatorThe Del Operator
ˆ ˆ ˆ
The Del Operator ( ) has the properties of a vector, and a
differential operator:
in Cartesian coordinatesx y za a ax y z
The del operator is important to us since it providesThe del operator is important to us since it providesa number of indications as to how vector and scalara number of indications as to how vector and scalarfunctions vary with position. It shows up in the gradient,functions vary with position. It shows up in the gradient,curl, divergence, and Laplacian.curl, divergence, and Laplacian.
GradientGradientThe gradient of a scalar function of position results in aThe gradient of a scalar function of position results in avector that points in the direction of greatest increase forvector that points in the direction of greatest increase forthat function. The magnitude of the gradient indicates howthat function. The magnitude of the gradient indicates howquickly that function changes with position.quickly that function changes with position.
ExampleExample
If the temperature in a room is
in what direction is the temperature increasing most rapidly, and
what is the rate of change with distance
( , , ) (65 0.
in that direction?
Solut
1 0.25
ion:
. )0 5T x y z x y z C
T
ˆ ˆ /
Rate of cha
ˆ(0.1 0.25 0.5 )
0.5n /7ge
x y z C m(x, y,z) a a a
T(x, Cy,z) m
If is a scalar function of position in Cartesian Coordinates
then ˆ ˆ ˆx y z
Φ(x, y,z)
Φ(x, y,z) Φ(x, y,z) Φ(x, y,z)Φ(x, y,z) a a a
x y z
Dr. Kent Chamberlin 2
GradientGradient (2)(2)
The normal to any surface defined byThe normal to any surface defined by f(x,y,z)=constant isconstant isgiven by the gradient ofgiven by the gradient of f(x,y,z)
2 2 2
2 2 2
Example: find the unit normal to the sphere
in Cartesian coordinates.
9
( , , )
ˆ ˆ ˆ( , , ) 2 2 2
ˆ ˆ2 2 2( ,
In this case ,
so , and the unit normal is
, )
( , ,
:
)
x y z
x y
x y z
f x y z x y z
f x y z xa ya za
xa ya zf x y z
f x y z
2 2 2 2 2 2
2 2 2since on the spherical surfac
ˆ ˆ ˆ ˆ ˆ ˆ ˆ
3
9 e
4 4 4
z x y z x y za xa ya za xa ya za
x y z x y z
x y z
CurlCurly
x screen.theintopoints:rulehand
righttheObeysrotation.maximum
inresultwillthatnorientatioaxiswheel
-paddlethedefiningvectorainresults
thenvelocity,fluidrepresentsIf
F
FF
In Cartesian coordinatesIn Cartesian coordinates
zyx
zyx
FFFzyx
aaa
F
ˆˆˆ
Dr. Kent Chamberlin 3
How the Curl WorksHow the Curl Worksy
x
Consider only the z-component of curl, whichcorresponds to the paddlewheel oriented asshown. That wheel will turn only if the xcomponent of the vector changes with y, orthe y component changes with x. The zcomponent of the curl is given by:
y
F
x
F
FFFzyx
aaaxy
zyx
zyx
ˆˆˆ
Curl ExamplesCurl Examples
Definition of the curlDefinition of the curl
zayF ˆ2
xayF ˆ2
y
x
t
HE
E
)(tH
max where is the surface
enclosed by the closed contour and is the unit
normal to that surface. The curl is a
ˆl
measure of
ro
im
0ˆ
tation per unit area.
n
S
n
a A dl
A ss s
a
Dr. Kent Chamberlin 4
DivergenceDivergenceDivergence is a measure of compression or decompressionof a field. It indicates the field leaving a volume element.
Because the divergence is a differential operator, it tells ussomething about the vector field at a particular point in space.
If the divergence of a vector field is positive throughout aregion, it indicates that more field is leaving that region thanentering.
The divergence can only be performed on a vector
ˆ ˆ ˆ ˆ ˆ ˆis a scalar
= in Cartesian coordinates
x y z x x y y z z
y
F
x z
F a a a F a F a F ax y z
FF F
x y z
How Divergence WorksHow Divergence Works
yy
VV y
y
yV
y
Consider the volume of gas leaving each faceConsider the volume of gas leaving each face
msec
Consider the net volume leaving the differential volume element
shown if gas velocity ˆ ˆ ˆx x y y z z
V V a V a V a
m( ) ( ) ( )3m( )
c es ce sLeft Face: - m m
y yV x z V x z
3m( )sec
Right Face: yy
VV y x z
y
x
y
z
Dr. Kent Chamberlin 5
How Divergence WorksHow Divergence Works (2)(2)Rear Face: ( )
3msecx
V y z
Front Face: ( )3m
secx
x
VV x y z
x
Bottom Face: ( )3m
seczV x y
Top Face: ( )3m
secz
z
VV z x y
z
The net loss per unit volume is (summing the above)The net loss per unit volume is (summing the above)
1sec
yx z
yx z
VV Vx y z x y z x y z
x y zx y z
VV VV
x y z
Divergence ExamplesDivergence Examples
z
y
,3
ˆ ˆ ˆIf gas velocity
is the region in compression or
decompression?
Since decompression
is the same everywhere.
V
V xa ya zax y z
2 ˆ
2
If gas velocity is the region in
compression or decompression?
Since decompression
increases with
yV y a
V y
y
Dr. Kent Chamberlin 6
Using Divergence in an EquationUsing Divergence in an Equation
As air leaves a region, its pressure decreases
As heat leaves a region, it cools off
As current leaves a region, its chargedensity decreases
If a region contains an electric charge density,an electric field will emanate from that region
If is air velocity, is air pressure, and is a constant, thenV k
V kt
If is heat flow, is temperature, and is a constant, thenH T k
TH k
t
If is current density, and is charge density, thenv ev
evv
J
Jt
If is electric field, and is charge density, thenev
ev
E
E
Definition of DivergenceDefinition of Divergence
lim
0
V dsV
v v
A measure of compression or decompression of aA measure of compression or decompression of afield. Indicates the field leaving per unit volume.field. Indicates the field leaving per unit volume.
max
ˆlim
0
n
S
a A dl
As s
Now, reconsider the definition of the CurlNow, reconsider the definition of the Curl