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Fields Lec 2
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Frequency Spectrum
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Coordinate systems
• EM fields vary in space and time
– need to be able to uniquely describe all points in space
• This may be done with curvilinear coordinate
systems
– orthogonal coordinate systems (i.e. coordinates are
perpendicular) are easiest to work with, and may
simplify problem solving!!
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Scalar- Magnitude only (e.g. charge, energy,
temperature).
Vector – Both magnitude and direction (e.g.
velocity, electric field intensity).
Scalar and Vector
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Cartesian coordinates
Independent variables: x y z
Ranges of independent variables?
Base unit vectors:
Represent vector as
A=Axax+ Ayay + Azaz
(Ax,Ay,Az)
Point is intersection of
3 planes
x
a y
a z
a
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Circular cylindrical coordinates
z
a a az
Independent variables:
Ranges of independent variables?
Base unit vectors:
The base vectors change their orientation at
different points in the system.
Point is intersections
of cylindrical surface,
plane and half-plane
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Circular cylindrical coordinates
Base vectors are mutually orthogonal
is in radians, not degrees
Represent vector A as:
A=Aa+ Aa + Azaz
(A,A,Az)
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Spherical coordinates
Independent variables:
Ranges of independent variables?
Base unit vectors: aaa
r
r
Point is intersection of a spherical
surface, cone and half-plane
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Spherical coordinates
Base vectors are mutually orthogonal
, are in radians, not degrees
Represent vector A as:
A=Arar+ Aa + Aa
(Ar, A, A)
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Why we need vector integration
Maxwell’s equations
emf = E dl
d
dt
B ds
c s
D ds dvvvs
B ds
s
0
+
ssc
sdE
dt
d
sdJldH
We need to define
the lines, surfaces
and volumes for
integration, and to
express the
associated dl, ds,
dv appropriately.
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Differential length – used in line
integrals
sc
sdJldH
We want to move along contour c (closed path) in small steps (dl).
In Cartesian coordinates, our contour could have components in x, y and z
directions the most general form of dl should have small changes in each
base vector direction
For example,
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Cartesian coordinates
To define differential length, area and volume,
we use a basic cube:
-small changes in x, y and z (dx, dy, dz)
• Differential volume:
-Volume of the cube: dv = dx dy dz
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dl
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For example,
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Differential normal area
- used in surface integrals
The surface is enclosed by the contour, c (assume that c has changes in 2
coordinates – e.g. x and y)
It has both magnitude (small) and direction (normal to surface).
sc
sdJldH
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dS
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Cylindrical coordinates
In Cartesian coordinates, we formed the cube by
moving the three defining planes incrementally
In cylindrical coordinates, we consider the shape
formed by expanding or moving the three
defining planes incrementally
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Spherical coordinates
Again, consider “moving” the three defining
planes incrementally
ddrdrdv sin2
aorasinorasin=s 2
rdrddrdrddrd r
asinaa=l
drrddrd r ++
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Summary
To perform vector integration (line, surface and volume),
we need to define appropriate differential elements
(length, surface and volume)
dl ds dv
Cart.
Cylind.
Spher.
aaa=l
zyx dzdydxd ++
aaa=l
zdzddd ++
asin
aa=l
dr
rddrd r ++
a
a
a
z
y
x
dxdy
dxdz
dydz
a
a
a
zdd
dzd
dzd
a
asin
asin2
rdrd
drdr
ddr r
dxdydzdv
dzdddv
ddrdrdv sin2
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Differential length – used in line
integrals
sc
sdJldH
We want to move along contour c (closed path) in small steps (dl).
In Cartesian coordinates, our contour could have components in x, y and z
directions the most general form of dl should have small changes in each
base vector direction
For example,
c
b
a
d dlcos|A|lA
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Line integrals
Line integral:
Integral of tangential component of A along curve C
From points a to b:
Closed path, L:
(also called circulation of A around L) (slide 88)
c dlA
L
dlA
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Surface integral
Surface integral:
flux of F through surface s
or, sum of components of F normal to the surface
(direction of s is perpendicular to surface)
Closed surface (calculate net outward flux of F)
s dsF
s
dsF
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n F
S
F
S
.
n
n
F
Flux parallel to area,
total flux crossing area
is zero
Flux perpendicular to
area
Total flux = F cos s
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Volume integral
Volume integral:
Integral of scalar over volume
In all cases, meaning of integral depends on the
quantity that we are integrating.
dvv
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Why we need vector
differentiation
Maxwell’s equations (differential form)
D v
B 0
+ +
H J E
E
t
E
B
t
We need to take the divergence
and curl (two kinds of
derivatives) of vector fields!
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Del operator
Vector differential operator ()
Definition:
Operates on a function
zyx a
z
a
y
a
x
++
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Del operator – cylindrical and
spherical coordinates
za
z
a
1
a
++
a
sin
1
a
1
a
r
r
rr
++
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Gradient
Definition: the gradient of a scalar field, V, is a
vector that represents both the magnitude and
direction of the maximum space rate of
increase of V
aaa
zyx
z
V
y
V
x
V
V
++
aa
1
a
z
z
VVV
V
++
a
sin
1
a
1
a
V
r
V
rr
V
V r ++
Example:
A scalar, time independent temperature field given in a region of space by
T= 3x + 2xyz - z2 – 2
1. Determine a unit vector normal to the isotherms (constant temp surface)
at the point (0,1,2)
2. Find the maximum rate of change of temperature at the same point
Solution:
The grad T is a vector perpendicular to the isotherms. A unit vector in this
direction is given by
65
4
65
7
65
4
)22(2)23(
z
x
zx
zyx
zyx
zxyxzyz
z
T
y
T
x
T
a
a
aa
n
aaa
aaa
n
+++
+
+
7
TT
T
2.The max. rate of temp
change equals the magnitude
of gradT. Maximum rate of
temp change at (0,1,2)is =sqrt
(65)
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Important points about gradient
Gradient vector is perpendicular to a surface of
constant value of the scalar field. This property
can be used to find a unit vector normal to a surface.
Magnitude of the gradient vector describes how
strongly the field varies with position, while the
direction of the gradient vector is in the
direction in which the scalar field increases most
rapidly.
If A= V then V is the scalar potential of A.
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Divergence of a vector
Divergence of A at given point P is a measure of the outflow of
flux from a small closed surface per unit volume as the volume
shrinks about P to zero (an operator that measures a vector
field’s tendency to originate from or converge upon a given
point P)
vv
s
0
sdA
limA
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The divergnece of a three dimensional vector field
is the extent to which the vector field flow behaves
like a source or a sink at a given point.
A vector field denoting the velocity of air
expanding as it is heated positive divergence.
At a given point, the divergence of A is just a
singular number representing how much the flow is
expanding at that point.
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Flux in =Flux out
No source or sink
Flux out Flux out
Flux out >Flux in
Source case
Flux out < Flux in
Sink case
Flux out
s
s
s
If C = x ax C = a r find .C
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Divergence of vector E in the three
coordinate systems
+
+
zy
zyx EE
x
E
E
+
+
z
zEE1E1
E
+
+
E
sin
1Esin
sin
1E1
E
2
2
rrr
r
r
r
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Curl
Definition: curl of A is a rotational vector
magnitude - maximum net circulation of A per unit area as
the area tends to zero
direction – normal to direction of area (area oriented such
that circulation is maximum)
curl orA A
max0
)lim( n
L
S
a
S
ldA
A
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Curl describes the properties of a field that cause rotation, and has both a magnitude and a direction. To
get a sense of curl, hold your arms straight out, palms facing forward and pretend that you are in a
vector field
a) uniform vector field b) non-uniform vector field causes rotation
1. The field is uniform (part a of figure): your palms are experiencing the same amount of force. The
field may “push” you forward or backwards but ignore this part for now.
2. The field is non-uniform (part b of figure): the left palm experiences greater force than the right,
causing you to rotate. Point the fingers of your right hand in the direction of rotation. Your thumb
points in the direction of the curl. The magnitude of the curl is determined by the difference between
forces on your left and right palms (i.e. how quickly you rotate)
3. The field is non-uniform: the right palm experiences greater force than the left, causing a different
rotation. Point the fingers of your right hand in the direction of rotation. In which direction is the curl?
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Classification of vector fields
A vector field is uniquely characterized by divergence
AND curl
Solenoidal (divergenceless)field: divergence is zero
No sources or sinks of flux
e.g. magnetic fields
divergence of the curl of a vector field = 0
a solenoidal field may be expressed as the curl of another vector
field
0 A
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Classification of vector fields
Irrotational field: curl is zero
Line integral of A is independent of chosen path
conservative field
E.g. electrostatic field
Gradient is irrotational can express irrotational field
as gradient of scalar field
A A dl dsc S
( )
0 A
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Summary
GRAD computes the gradient of a scalar function
It finds the gradient, the slope, how fast you change in
any given direction.
DIV computes the divergence of a vector.
It finds how much “stuff” is leaving a point in space.
CURL computes the rotational aspects of a vector
function.
Assume the vector field: B = r cos ar + sin a
Find:
over the surface of the semicircular
contourarsemicircultheoverdB
c
.
s
dsB .
dl = dr ar + d a + dz az
