Equipped with our power calculus equation, we can now derive the field created by rings and coils.
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:
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.
However, this vector is at an angle θ to the z axis. Thus the component of the field produced by dl in the z -axis 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 = =|
|B z =|
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:
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.
|B = (cosθ 1 - cosθ 2)|
From the above equation we can generate an expression for the field of a solenoid infinite in length. In an infinite solenoid there is a uniform magnetic field in the direction of the axis, given by:
|B = (cos 0 - cosΠ) =|
The study of these complex wire shapes concludes our study of the sources of magnetic fields. In the next SparkNote in the series on magnetic forces and fields we will take a more theoretical approach to magnetism, describing some of the properties of all magnetic fields.