Notes 2 Spring 2005.doc
Scalar Fields and Vector Fields
A voltage as a function of time may be written as
( )
t
v
.
This is an example of a function of one
variable.
In electromagnetics this, as well as other quantities of interest, are also often functions
of time
and
space.
In the Cartesian coordinate system we might write
( )
t
,
z
,
y
,
x
v
v
→
The fact that this quantity is now a function of three dimensional space makes this a
field
.
The
fact that this quantity is a scalar (there is no direction associated with it) as opposed to a vector,
makes this a
scalar field
.
Another good example of a scalar field is the temperature at different
locations in this room.
Since vector quantities are often of interest in physics and engineering (force, velocity,
momentum, current density, electric and magnetic fields, etc. …) it is reasonable to want to
consider
vector fields
, which are similar to scalar fields except each point in space not only has a
magnitude, but also has a direction associated with it, i.e.,
( )
t
,
z
,
y
,
x
E
E
G
G
→
G
Choosing the symbol
E
for this example is appropriate because we are thinking about the
electric field which is a vector quantity with both a magnitude and direction.
Staying with the
Cartesian coordinate system we can decompose this vector field into its components like this,
()
( ) (
)(
z
ˆ
t
,
z
,
y
,
x
y
ˆ
t
,
z
,
y
,
x
x
ˆ
t
,
z
,
y
,
x
t
,
z
,
y
,
x
z
y
x
E
E
E
E
)
+
+
=
G
The three components of
,
,
,
are themselves, scalar fields, and so we can say that to
describe the information in a vector field we need to keep track of three scalar fields.
,
, and
are
unit vectors
(of length one) and balance the vector nature of the left hand side of the
equation with the right hand side.
E
G
x
E
y
E
z
E
x
ˆ
y
ˆ
z
ˆ
Taking derivatives of fields:
The del operator, the gradient, the divergence
and the curl.
We can take the derivative with respect to time of either a scalar or vector field:
()
[]
( )
t
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dt
d
∂
∂
=
v
v
()
()
( )
( )
z
ˆ
t
t
,
z
,
y
,
x
y
ˆ
t
t
,
z
,
y
,
x
x
ˆ
t
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dt
d
z
y
x
∂
∂
+
∂
∂
+
∂
∂
=
E
E
E
E
G
1
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We can also take the derivative with respect to any of the three Cartesian
*
directions in space:
()
[]
( )
x
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dx
d
∂
∂
=
v
v
()
[]
()
( )
( )
z
ˆ
z
t
,
z
,
y
,
x
y
ˆ
z
t
,
z
,
y
,
x
x
ˆ
z
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dz
d
z
y
x
∂
∂
+
∂
∂
+
∂
∂
=
E
E
E
E
G
see example 1 below
Remember that a derivative gives us information about how a function is
changing
with respect
to some variable (often a direction or time).
For a
scalar field
, as shown above, we can take the
derivative with respect to one direction, or we can
create
an expression (and call it the gradient
of the scalar field) that shows how the scalar field is changing in
each
of the three directions by
adding together the derivatives in each direction along and attaching the corresponding unit
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 Fall '10
 Czarnecki
 Electromagnet, Volt, Vector field

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