In order to establish some properties of the magnetic field, we must review some
of the principles of vector calculus. These principles will be our guidance in
the next
section.
Divergence of a Vector Field and Gauss' Theorem
Consider a three dimensional vector field defined by
F
= (P, Q, R)
, where
P
,
Q
and
R
are all functions of
x
,
y
and
z
. A typical vector field, for example, would be
F
= (2x, xy, z
^{2}
x)
. The divergence of this vector field
is defined as:
diverge
Thus the divergence is the sum of the partial differentials of the three
functions that constitute the field. The divergence is a function, not a field,
and is defined uniquely at each point by a scalar. Speaking physically, the
divergence of a vector field at a given point measures whether there is a net
flow toward or away the point. It is often useful to make the analogy comparing
a vector field to a moving body of water. A nonzero divergence indicates that
at some point water is introduced or taken away from the system (a spring or a
sinkhole). Recall from electric forces and fields that the divergence of an
electric field at a given point is nonzero only if there is some charge density
at that point. Point charges cause divergence, as they are a "source" of field
lines.
Divergence is mathematically significant because it allows us to relate volume
integrals and surface integrals, through Gauss' Theorem. Given a closed
surface that encompasses a certain volume, this theorem states that:
where the left side is a surface integral over a and the right side is a volume
integral. We don't really deal with volume integrals in electricity and
magnetism, so some of this theorem is irrelevant. However, when the divergence
of a vector field is zero, this equation tells us that the integral through any
surface in the field must also be zero.
The Curl of a Vector Field and Stokes' Theorem
The second major concept from vector calculus that applies to magnetic fields is
that of the curl of a vector function. Take again our vector field
F
= (P, Q, R)
. The curl of this vector field is defined as:
Clearly this equation is a bit more complicated, but it gives us a lot more
information. The curl, unlike the divergence, is itself a vector field, defined
by a single vector at each point. Physically speaking, curl measures the
rotational motion of a vector field. Again using our water analogy, a nonzero
curl indicates an eddy or a whirlpool. At a given point in the field, the curl
at that point tells us the axis of rotation of the field about that point. If
the curl is zero, there is no axis of rotation, and thus no circular motion.
Unlike magnetic fields, electric fields never have curls. Recall that the line
integral over any closed loop in an electric field is zero, implying that the
field cannot "curve" around, as a field with a nonzero curl would.
Just as Gauss' Theorem relates surface integrals and volume integrals using
divergence, Stokes' Theorem relates surface integrals and line integrals
using curl. Given a closed curve that encompasses a surface,
where the left side is a line integral and the right side is a surface integral.
Again, we pay special attention to the special case in which the curl is zero.
In this case, the integral of the field around any closed loop is zero.
Electric fields have this property.
It is incredibly important when using Gauss' and Stokes' Theorem to remember
that one must deal with closed surfaces (with Gauss' Theorem) or closed loops
(with Stokes' Theorem). Otherwise the equations are not applicable.
Having established the concepts of curl and divergence, and related them to
vector calculus through our two theorems, we can apply these concepts to
magnetic fields.