Our story thus far has been that charges create electric fields and that moving charges, or currents, create magnetic fields. But consider FIGURE 31.1a, where Brittney, carrying charge q, runs past Alec with velocity $\vec{v}$. Alec sees a moving charge, and he knows that this charge creates a magnetic field. But from Brittney’s perspective, the charge is at rest. Stationary charges don’t create magnetic fields, so Brittney claims that the magnetic field is zero. Is there, or is there not, a magnetic field?

FIGURE 31.1 Brittney carries a charge past Alec.

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Or what about the situation in FIGURE 31.1b? Now Brittney is carrying the charge through a magnetic field that Alec has created. Alec sees a charge moving in a magnetic field, so he knows there’s a force $\vec{F}=q\vec{v}\times \vec{B}$ on the charge. But for Brittney the charge is still at rest. Stationary charges don’t experience magnetic forces, so Brittney claims that $\vec{F}=\vec{0}$

Now, we may be a bit uncertain about magnetic fields, but surely there can be no disagreement over forces. After all, forces cause observable and measurable effects, so Alec and Brittney should be able to agree on whether or not the charge experiences a force. Further, if Brittney runs with constant velocity, then both Alec and Brittney are in inertial reference frames. You learned in Chapter 4 that these are the reference frames in which Newton’s laws are valid, so we can’t say that there’s anything abnormal or unusual about Alec’s and Brittney’s observations.

This paradox has arisen because magnetic fields and forces depend on velocity, but we haven’t looked at the issue of velocity with respect to what or velocity as measured by whom. The resolution of this paradox will lead us to the conclusion that $\vec{E}$ and $\vec{B}$ are not, as we’ve been assuming, separate and independent entities. They are closely intertwined.