Week 7: Sources of the Magnetic Field 231
charges we didn’t find any. Yet magnetic fields exist in abundance; otherwise how could pictures
and newspaper articles ever be stuck to our refrigerator doors?
When we go to search for sources of these magnetic fields, we find that they all have something
in common. They are for the most part produced bymoving electrical charges^64.
Time to go into a laboratory and perform experiments. These experiments are actually rather
difficult to do for single charges, so (as we’ll see in a moment) the original experiments more com-
monly involved electricalcurrentsmade up of many moving charges, but we’ll find it a bit more
useful to start at the microscopic end and then average to find the macroscopic rule. From them we
learn that:
- The magnetic field of a moving (electrical) chargeqis proportional to the charge.
- The magnetic field of a moving charge is inversely proportional to thedistance from the charge
squared, just as was the case of the electrostatic field. - Thedirectionof the magnetic field is that it formsloopsaround the direction of motion of the
charge, in the direction the fingers of your right hand curl aroundyour thumb if you line your
thumb up along with the direction of motion of the positive charge. - For any given value ofq,vandr, the field strength is proportional to sin(θ) whereθis the
angle between the direction of~vand the direction of~r.
These empirical observations can be summarized in the following formula^65 :B~=kmq(~v׈r)
r^2=
μ 0
4 πq(~v×rˆ)
r^2(497)
so that the magnetic field has themagnitude:
B=kmqvsin(θ)1
r^2(498)
The geometry of the field lines is show in figure 77. Note that if you let the thumb of your right
hand line up with the direction of motion of the charge, your fingers will curl around your thumb in
the same sense that the field lines are directed around the velocity vector of the charge.
Finite Field Propagation Speed forEandB.
This rule is simple enough, and isalmostthe rule you will learn in much more advanced courses in
electrodynamics than this one. The only thing we are leaving out (that is, of course, very important)
is that neither the electric nor the magnetic field appears instantaneously in all space. When one of
their sources is “turned on” by a suitable rearrangement or motionof charges, the fields propagate
outward from the charge at the speed of light, establishing its valueat a point at a lagged timeafter
the charge or current appears at any given point.
(^64) Reality is slightly more complicated than that, however. For one thing, point-like electrons have a magnetic dipole
moment due to their spin even though it is difficult to describea point-like object as “moving”. For another, as we
shall see, changing electric fields make magnetic fields evenin the absence of moving electrical charges. Still, we will
be able to understandbothphenomena in terms of moving electrical charge at least at first, and later can visualize
and bootstrap an understanding of point-like magnetic dipoles as a kind of limit of rotating macroscopic charge.
(^65) This equation was originally obtained by Oliver Heaviside in 1888, long after the Biot-Savart Law forcurrents
(next) was discovered, but it is in some sense more fundamental. However, it is alsoflawed. For one thing, it lacks
retardation– the field emitted from the moving charge propagates at the speed of light and is not instantaneous. This
is not a problem with the Biot-Savart Law because it is amagnetostaticlaw valid for more or less continuous and
steady currents. It is worth keeping in mind while reading this chapter that with the exception of Gauss’s Law for
Magnetism, every fundamental equation taught herein isnot quite trueand is presented as they are in elementary
courses to help bootstrap Maxwell’s Equations as they are eventually (correctly) written.