Simple Nature - Light and Matter

the two wires of example 8: current goes out one wire, but comesback through the other. Since the field really falls off as R −2 , wehave an energy density that varies as R −4 , which does not give infinitywhen integrated out to infinity. (There is still an infinity atR = 0, but this doesn’t occur for a real wire, which has a finitediameter.)Still, one might worry about the physical implications of thesingle-wire result. For instance, suppose we turn on an electrongun, like the one in a TV tube. It takes perhaps a microsecond forthe beam to progress across the tube. After it hits the other sideof the tube, a return current is established, but at least for the firstmicrosecond, we have only a single current, not two. Do we haveinfinite energy in the resulting magnetic field? No. It takes time forelectric and magnetic field disturbances to travel outward throughspace, so during that microsecond, the field spreads only to somefinite value of R, not R = ∞.This reminds us of an important fact about our study of magnetismso far: we have only been considering situations where thecurrents and magnetic fields are constant over time. The equationB = 2kI/c 2 R was derived under this assumption. This equation isonly valid if we assume the current has been established and flowingsteadily for a long time, and if we are talking about the field ata point in space at which the field has been established for a longtime. The generalization to time-varying fields is nontrivial, andqualitatively new effects will crop up. We have already seen oneexample of this on page 598, where we inferred that an inductor’stime-varying magnetic field creates an electric field — an electricfield which is not created by any charges anywhere. Effects likethese will be discussed in section 11.5A sheet of currentThere is a saying that in computer science, there are only threenice numbers: zero, one, and however many you please. In otherwords, computer software shouldn’t have arbitrary limitations likea maximum of 16 open files, or 256 e-mail messages per mailbox.When superposing the fields of long, straight wires, the really interestingcases are one wire, two wires, and infinitely many wires. Withan infinite number of wires, each carrying an infinitesimal current,we can create sheets of current, as in figure d. Such a sheet has acertain amount of current per unit length, which we notate η (Greekletter eta). The setup is similar to example 8, except that all thecurrents are in the same direction, and instead of adding up twofields, we add up an infinite number of them by doing an integral.d / A sheet of charge.Section 11.2 Magnetic Fields by Superposition 663