Equipped with our power calculus equation, we can now derive the field created
by rings and coils.
Field of a Single Ring
Consider a single wire wrapped in a circle, and carrying a current. From our
second right hand rule, we can describe qualitatively the magnetic field created
by the current. Shown below is such a field:
Figure %: The field created by a ring. If the ring lies in the xy plane, then
the field lines point in the positive z direction
It is clear that on the axis of the ring, the field lines point straight up,
perpendicular to the plane of the ring. Notice the similarity between the field
of a ring and that of a magnet. This is not a coincidence, and can be described
using atomic theory of ferromagnetic materials.
We can also determine the strength of this field on the axis. Consider a point
on the axis, elevated a distance z from the plane of a ring with radius b, shown
below.
Figure %: A point of the axis of the ring, shown with relevant distances and
angles to an element of length, dl.
Fortunately,
dl and
are perpendicular in this case, greatly
simplifying our equation for
dB:
dB =
However, this vector is at an angle
θ to the
z axis. Thus the component
of the field produced by
dl in the
zaxis is given by:
The geometry used to get this equation can be seen from the .
Now we integrate this expression over the entire circle. Notice, however, that
dl = 2Πb, or simply the circumference of the circle. Thus:
B_{z} = = 

This equation applies to any point on the axis of the ring. To find the field
at the center of the ring, we simply plug in
z = 0:
B_{z} = 

Thus we have a set of equations for the field of a ring. Though the derivation
required calculus, and may not be useful, it allowed us to get some experience
using our complex equation from the last section. Next we
stack a number of rings on top of each other, and analyze the resultant field.
Field of a Solenoid
In many instances a wire is coiled in a helical pattern to create a
cylindrically shaped object known as a solenoid. These objects are
frequently used in magnetic experiments, as they create an almost uniform field
inside the cylinder. The solenoid can be seen as the superposition of a large
number of rings, one on top of the other. Shown below is a typical solenoid,
with its field lines:
Figure %: A solenoid, shown with some field lines
The field has a similar shape as a ring, but appears more "stretched", a result
of the cylindrical shape of the object.
We can use the same method to find the magnitude of the magnetic field on the
axis of the solenoid that we did with the ring. However, the calculus is long
and complicated and, since we have already gone through the process, we will
simply state the equations.
Consider a solenoid with n turns per centimeter, carrying a current I, shown
below.
Figure %: The inside of a solenoid, shown with a point P on the axis of the
solenoid
The field at point
P is given by:
B = (cosθ_{1}  cosθ_{2}) 

where
θ_{1} and
θ_{2} are the angles between vertical and the lines
from
P to the edge of the solenoid, as shown in the figure. Analyzing this
equation we see that the longer the solenoid, the greater the magnitude of the
magnetic field.