Using vector calculus, we can generate some properties of any magnetic field,
independent of the particular source of the field.
Line Integrals of Magnetic Fields
Recall that while studying electric fields we
established that the surface integral through any closed surface in the field
was equal to
4Π
times the total charge enclosed by the surface. We wish to
develop a similar property for magnetic fields. For magnetic fields, however,
we do not use a closed surface, but a closed loop. Consider a closed circular
loop of radius
r
about a straight wire carrying a current
I
, as shown below.
A closed path around a straight wire
What is the line integral around this closed loop? We have chosen a path with
constant radius, so the magnetic field at every point on the path is the same:
B =
. In addition, the total length of the path is simply the
circumference of the circle:
l = 2Πr
. Thus, because the field is constant on
the path, the line integral is simply:
lineintegral
This equation, called Ampere's Law, is quite convenient. We have generated an equation for the line
integral of the magnetic field, independent of the position relative to the
source. In fact, this equation is valid for any closed loop around the wire,
not just a circular one (see problems).
@@Equation @@ can be generalized for any number of wires carrying
any number of currents in any direction. We won't go through the derivation,
but will simply state the general equation.
B·ds = × total current enclosed by path


Note that the path need not be circular or perpendicular to the wires. The
figure below shows a configuration of a closed path around a number of wires:
Figure %: A closed path enclosing 4 wires
The line integral around the circle in the figure is equal to
(I
_{1} + I
_{2}  I
_{3}  I
_{4})
. Notice that the two wires pointing downwards
are subtracted, since their field points in the opposite direction from the
curve.
This equation, similar to the surface integral equation for electric fields, is
powerful and allows us to greatly simplify many physical situations.
The Curl of a Magnetic Field
From this equation, we can generate an expression for the curl of
a magnetic field. Stokes' Theorem states that:
B·
ds =
curl
B·
da
We have already established that
B·ds =
. Thus:
curl B·da =
To remove the integral from this equation we include the concept of current
density,
J
. Recall that
I =
J·da
. Substituting this into our
equation, we find that
Clearly, then:
=


Thus the curl of a magnetic field at any point is equal to the current density
at that point. This is the simplest statement relating the magnetic field and
moving charges. It is mathematically equivalent to the line integral equation
we developed before, but is easier to work with in a theoretical sense.
The Divergence of the Magnetic Field
Recall that the divergence of the electric field was equal to the total charge
density at a given point. We have already examined qualitatively that there is
no such thing as magnetic charge. All magnetic fields are, in essence, created
by moving charges, not by static ones. Thus, because there are no magnetic
charges, there is no divergence in a magnetic field:
= 0


This fact remains true for any point in any magnetic field. Our expressions for
divergence and curl of a magnetic field are sufficient to describe uniquely any
magnetic field from the current density in the field. The equations for
divergence and curl are extremely powerful; taken together with the equations
for the divergence and curl for the electric field, they are said to encompass
mathematically the entire study of electricity and magnetism.