40
DIFFERENTIAL AND INTEGRAL
OPERATIONS
which you will prove at the end of the chapter. Write
v
as
(2.100)
-
V=VXA-V@.
Then we can write:
-
v
.v
=
-V2@
-
v
x
v
=
v
x
v
x
K.
ii.v=
ii-
(VXA-ED),
(2.101)
Since the divergence, curl, and normal component are all specified if and
@
are
specified, Helmholtz’s Theorem says
v
is also uniquely specified. Notice the
contribution to
v
that comes from has no divergence since
v
*
(v
X
;Ir>
=
0.
This
is called the rotational or solenoidal part
of
the field and is called the vector
potential. The portion of
X
v@
=
0.
This is called the irrotational part of the field and
@
is
called the scalar potential.
which arises from
@
has
no
curl, because
EXERCISES
FOR
CHAPTEX
2
1.
Consider the two-dimensional scalar potential function
CP(x,y)
=
x3
-
3y2x.
Make a plot
of
the equipotential contours for three positive and three negative
values for
CP.
Find
v@
.
Show that the
v@
field lines are given by setting
3x2y
-
y3
equal to a series
of constants.
Plot six representative
V@
field lines. Be sure to indicate the direction
of
the
field and comment on its magnitude.
Find
V
*
VCP,
the divergence
of
the
v@
vector field, and show that
your
field lines
of
part (d) agree with
this
divergence.
electric dipole
p
is located at the origin
of
a Cartesian system. This dipole
__
creates an electric potential field
@@)
given
by
where
F
is
the position vector,
F
=
ni4.
Let
(a)
Sketch the equipotential lines,
@
=
constant.
(b)
Find the electric field,
a
=
-v@.
(c)
Sketch the electric field lines.
=
p&.
