Notes 2
Fall 2004.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 the atmosphere.
A hot air balloonist, for instance, traveling between Baton Rouge
and New Orleans might be concerned about the temperatures at different altitudes
( )
z
, at
different points,
along his route.
(
y
x
,
)
Since vector quantities are often of interest (force, velocity, momentum, current density, electric
and magnetic fields …) it is reasonable to want to consider
vector fields,
( )
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.
Differentiating scalar and vector fields:
()
[]
( )
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentNotes 2
Fall 2004.doc
We can also take the derivative with respect to a particular direction in space:
For the scalar field,
()
[]
( )
x
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dx
d
∂
∂
=
v
v
For the vector field,
()
[]
()
( )
( )
z
ˆ
x
t
,
z
,
y
,
x
y
ˆ
x
t
,
z
,
y
,
x
x
ˆ
x
t
,
z
,
y
,
x
t
,
z
,
y
,
x
dx
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 (usually a direction or time).
For a scalar field, as shown in the example above
we can take the derivative with respect to one direction, or we can
create
an expression that
shows how the scalar field is changing in
each
of the three directions by adding together the
derivatives in each direction along with the unit vector for each term (which results in a vector
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Czarnecki
 Electromagnet, Volt, Vector field

Click to edit the document details