Question 7.9: An infinitely long straight wire carries a slowly varying cu...

In the quasistatic approximation, the magnetic field is (μ_0 I/2πs), and it circles around the wire. Like the B-field of a solenoid, E here runs parallel to the axis. For the rectangular “Amperian loop” in Fig. 7.27, Faraday’s law gives:

E(s)=\left[\frac{\mu _{0}}{2\pi}\frac{dI}{dt}\ln s+K \right]\hat{Z},          (7.20)

where K is a constant (that is to say, it is independent of s—it might still be a function of t). The actual value of K depends on the whole history of the function I (t)—we’ll see some examples in Chapter 10.
Equation 7.20 has the peculiar implication that E blows up as s goes to infinity. That can’t be true . . . What’s gone wrong? Answer: We have overstepped the limits of the quasistatic approximation. As we shall see in Chapter 9, electromagnetic “news” travels at the speed of light, and at large distances B depends not on the current now, but on the current as it was at some earlier time (indeed, a whole range of earlier times, since different points on the wire are different distances away). If τ is the time it takes I to change substantially, then the quasistatic approximation should hold only for

s\ll c\tau ,                            (7.21)

and hence Eq. 7.20 simply does not apply, at extremely large s.