W9_parallel_resonance.eps

Week 7: Sources of the Magnetic Field 235

r

r

r − r

dl

I

0 dB

0

Figure 80: The Biot-Savart Law in general coordinates.

Wow! That looks a lot more complicated than our integral expressions for the electrostatic field!
And so it is... we will have to work much harder to evaluate the magneticfield directly from a current
(distribution), and the need to do twisty integrals of directed differential vectors as they are worked
along a curveandformed into a cross product with a relative vector to the point of observation (in
some system of coordinates) will severely limit the problems we can solve analytically by just doing
sufficiently straightfoward integrals.

This is good news and bad news. From the student’s point of view, it means that things at the
intro level are relatively easy. If you learn to do all of the examples,well, that iscloseto all of the
examples one can do without being an integration god. On the other hand, it presages bad things
for the more advanced student, where sooner or later some of the more difficult problems must be
faced (at which point you will need to have made some progress on the road to calculus-deity).

For this course, we will cheerfully take the easy road and work through a nice set of relatively
simple examples that is, as promised, most of or close to what you cando, period, before things get
very complicated indeed.

7.4: Examples of Using the Biot-Savart Law to Find the Mag-

netic Field

Example 7.4.1: Magnetic Field of a Straight Wire Segment

I dx

r

θ

y

x

dB (out)

φ

dθ

θ 1 θ 2

x 1 x 2

Figure 81: The geometry and coordinates that make it simplest to evaluate the magnetic field of a
straight segment of wire carrying a currentI.

In figure 81 above, we see the geometry of a single, straight, segment of wire relative to an
arbitrary point in space. We wish to use the Biot-Savart Law to find the field of this wire at the
point on they-axis indicated. Note well that this is ageneral pointas they-axis itself is located at