Space‐Charge Flow with Unrestricted Variation of Current Density and Energy Range
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11.Potentials within the cavity are drawn in the positive direction to better indicate that the potential energy of the electrons is positive, and thus to avoid confusion in diagrams involving the sum of potential and (always positive) kinetic energies. Furthermore, it is easier to visualize an electron as slowing down as it moves up a potential barrier. For convenience in the calculations the electronic charge e as well as the potentials are taken as positive, but equations are written in the usual manner.
12.No problem is produced if other charge sheets happen to lie within the same interval as the sheet being considered. The effect is to round off the sharp‐pointed potential curves of these sheets, a condition which may approximate physical reality rather well since a charge sheet is, in reality, the mathematical representation of a layer of charge having finite thickness and with a rounded potential distribution through the layer.
13.As a sheet approaches or recedes from the spot where it is quite possible at the close of a time interval that may be greater than the quantity (see Fig. 4), and thus by Eq. (15) the indicated kinetic energy is negative. However, this physical unreality causes no difficulties in the computations. For the nonrelativistic case the acceleration, by Eq. (16), is independent of For the relativistic case a negative value for used in Eq. (20) results in a value for the calculated acceleration that is equal to the nonrelativistic value when occurs near the center of the increment, a slightly smaller value if it occurs beyond this position and slightly larger if closer, but the variation is inconsequential since it occurs near zero velocity and requires no correction.
14.A smaller number of distance intervals can be used when the potential curve is always relatively smooth, but a larger number is needed when the curve acquires a sharp break or peak in a region where sheets of a given energy tend to accumulate because of their total or partial reflection. A system could be devised for automatically dividing such a region into smaller intervals whenever necessary, rather than using the smaller intervals throughout the entire calculations.
15.See Ref. 9, pp. 118–122.
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