V.4 No 1

7

On gradient of potential function

Due to this, the dynamic gradient will be described by two projections  _r  and , and the equation for finding the gradient can be represented in the standard form

We can transform (14) to separate the derivatives by increments, as we did it in the case of stationary source. Then we yield

As we can see from (15), in case of moving source the gradient of scalar potential also divides into two summands. First of them is the standard expression for the gradient of scalar function in coordinates, and the second, just as in previous case, determines the time variation of potential. Of course, it would be hard in general case to substitute directly the second summand by vector potential, as we did it in case of pulsing source. Here we have to consider the conditions of model in each case separately, but for example for far field this second summand becomes much simpler, as we can neglect the tangential projection of  .

Actually, if the field source oscillations around the balance position have a harmonic pattern, in accordance with Fig. 4 we can write

where is the radiation wavelength.

It follows from (18) that in the far field, where  r >>   and if the derivative is limited, the expression in the right-hand part will be arbitrary small; this allows us to neglect it, in comparison with the radial component in (15). Factually this operation means that for the far field we neglect the distance from the points of trajectory of source motion to the studied point  P₀   - just what we did in [3], in transition from the integral solution of Laplace equation for vector potential to that analytical.

Noting (18), we come to the expression similar to that which we yielded for pulsing source,

which satisfies the condition (12).

For the near field, where we may not neglect the tangential component, it will be correct to write the connection between the scalar and vector potentials as

It is easy to check that the equation of relation (20) satisfies (12).