Mathematical Physics Vol 1

4.3 Examples of some fields

105

R Note that in the case when several points attract one point, the potential is

n ∑ i = 1 n ∑ i = 1

m i r i

U = U ( x , y , z )=

=

m i

p ( x − x i )

=

.

2 +( y − y

2 +( z − z

2

i )

i )

Newton’s force by which a homogeneous spherical shell attracts a material point

Consider now a homogeneous spherical shell, of radius R , which attracts a material point (Fig. 4.13). Let us first determine the force by which an infinitesimal part of the sphere (d S ) attracts the observed point d F = k 2 m 1 d m 2 r 2 r r . (4.104) Given that d m 2 = ρ d V = ρ · d · d S , for unit thickness d = 1 we obtain d m 2 = ρ d S . There is always one straight line through the observed point M and the center of the sphere O . In our case, let it be the z – axis, i.e. OM = z .

z M m = 1

Let us denote by r the distance between the point M and the center of the surface d S (the point where the force of attraction acts). This distance is, according to the figure (application of the cosine theorem) r 2 = R 2 + z 2 − 2 zR cos θ . (4.105) For the magnitude of the force we now ob tain

1

r

d S

q

R

O

k 2 m 1 ρ d S R 2 + z 2 − 2 zR cos θ .

d F =

(4.106)

Figure 4.13: Spherical shell - material point M .

Projecting this force on the z – axis, and taking into account the symmetry, we can notice that the projections on the normal direction, perpendicular to z , are mutually canceled, so only the Z – projection should be taken into account (see Fig. 4.14).

Figure 4.14: Projection of force on the z - axis.