Mathematical Physics Vol 1

4.3 Examples of some fields

107

Attraction of a point by a line body Let us determine the potential of a homogeneous line L , which coincides with the z – axis. As the observed point M and the given line L determine a plane, let us assume that plane is the xOz plane, see Fig. 4.15.

Figure 4.15: Attraction of a point by a line body.

Let us first determine the force by which the line attracts the point. The elementary force is

d z r 2

d z x 2 + z 2

| d F | = k

= k

(4.116)

.

R Note that we assumed that the force is proportional to the masses, and inversely proportional to the square of the distance. However, given that d m = ρ d V = ρ P d z and P = 1, we obtained that the force depends on d z , while we denoted all other constants with one letter – k . Further, similarly as in the previous example, due to the symmetry of the projections (see Fig. 4.15), once again only the X – projection needs to be determined d X = d F cos α = − k d z r 2 · x r . (4.117) The total projection is X = − Z + ∞ − ∞ k x r · d z r 2 . (4.118) Further, given that

z x ⇒

x cos 2 α

tg α =

z = x tg α ⇒ d z =

d α , x = const ,

(4.119)

and

x 3 cos 3 α

r 3 =( x 2 + z 2 ) 3 / 2 = x 3 ( 1 + tg 2 α ) 3 / 2 =

(4.120)

,

for X we obtain

X = − k Z

− π 2

cos 3 α x 3

x cos 2 α

d α .

x ·

(4.121)

π 2

+