Mathematical Physics Vol 1

4.3 Examples of some fields

103

4.3 Examples of some fields of interest for physics and engineering

We will now present some examples of potential fields that are of particular interest for physics and engineering. Attraction of two points in the field of gravitational force

Newton’s gravitational force is defined by the expression F = − γ mm 0 r 2 r 0 , (4.94) where m and m 0 are masses being at tracted, and γ is the universal gravita tional constant. In the previous expression we have in troduced the following notation (Fig. 4.11)

Figure 4.11: Attraction of two points.

r r

where r = −−→ M 0 M .

r 0 = (4.95) The question is, whether there exists a potential for a force defined in such a manner? As masses m and m 0 , aswell as γ , are constant, we can substitute them by a single constant, for example c , and we can then represent the force in the following form F = − c r 2 r 0 . (4.96) We have shown earlier (see p. 93) that if there exists a scalar function U , such that F = grad U , then the vector field F is potential. Thus, such a scalar function U = U ( r ) should be determined. Given that grad U = d U d r r 0 i F = − c r 2 r 0 from F = grad U we obtain d U d r = − c r 2 that is d U = − c r 2 d r and then U = 2 c r + c 1 . Based on these results, it can be concluded that the gravitational force, defined by (4.94), can be represented by F = grad U , (4.97) i.e. the force is potential, and its potential can be determined by the expression ,

C r

U =

(4.98)

.

This potential is also known in literature as Newton’s potential . Here, we assumed that 2 c = C and c 1 = const . = 0, which does not affect the generality of the previously derived expression. It has been shown that ∆ ( 1 / r )= 0.