Transient Magnetic Fields by Neil R. Sheeley

Author:Neil R. Sheeley
Language: eng
Format: epub, pdf
ISBN: 9783030402648
Publisher: Springer International Publishing

(10.40)

It is instructive to expand the expression for ΔA in Eq. (10.39) as a power series in 1∕ Δr. Including terms through order 1∕( Δr)3, we find that

(10.41)

This means that the radiative energy given by Eq. (10.38) becomes

(10.42)

consistent with the central contribution to the radiated energy that we obtained in the previous section, as given in Eq. (10.16). Thus, when Δr ≫ 2, the contribution from the middle of the transient varies as ( Δr)−3 and is proportional to the square of the “acceleration,” like the radiation from an accelerating charge in the non-relativistic limit (Leighton 1959).

Equation (10.41) also implies that H S − H 2 = −H″(y)∕{3( Δr)2}. Substituting the values of H″(y) = 24(y − 1∕2) and Δr = 2∕0.3, we obtain H S − H 2 = −0.18(y − 0.5), as the equation for the middle portion of the purple curve in the right panel of Fig. 10.4. So it is no coincidence that this section of the curve looks like a straight line. Of course, it would not be a straight line if the ramp profile, H″(y), were a non-linear function of y.

It is interesting to see how Eqs. (10.38) and (10.39) apply to some other current ramps. A simple example is the sudden turn-on that jumps from 0 to 1 at the midpoint of the interval. As the observation window encounters the jump, there are opposite contributions from regions above and below the jump. These contributions cancel when the window is centered on the jump, but for an arbitrary offset distance x, they give ΔA = x. Therefore, ( ΔA)2 = x 2 for x in the interval (− 1∕ Δr, +1∕ Δr) and ( ΔA)2 = 0 otherwise. In this case, the integral in Eq. (10.38) has the value (2∕3)( Δr)−3, and , as expected for a sudden onset.

The linear ramp provides another example. The running window sees no excess area from the middle part of the ramp, so we need only consider the ends. When the start of the ramp lies at a location x in the observation window, the excess area is ΔA = (x∕2)(2∕ Δr − x). Therefore, ( ΔA)2 = (x∕2)2 (2∕ Δr − x)2 for x in the range (0, 2∕ Δr) and ( ΔA)2 = 0 otherwise. For this range of integration, we obtain from the start of the ramp. An equal contribution is obtained from the end of the ramp, so that the total radiated energy is (6∕5)( Δr)−2 when Δr ≥ 2, in agreement with Eq. (9.​17). When Δr ≤ 2, the running window becomes wider than the ramp, and all three regions contribute to the radiated energy, giving , in agreement with Eq. (9.​18).

We can gain further insight by considering profiles that do not change rapidly on the interval (0, Δr). Near the start of a very long ramp with Δr ≫ 2, we may approximate H(x) by the first non-vanishing term in its power series expansion about x = 0:

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