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Sources of Magnetic Fields

Fields of Rings and Coils

Problems

Problems

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 x - y 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 z -axis is given by:

dB z = cosθ =

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.

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) =    

This is the magnitude of the uniform field inside the solenoid. The field outside an infinite solenoid is always zero.

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.

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