Tensors for Physics

7.1 Scalar Fields, Gradient 83

(iii) Spherical Symmetry
Now the case is considered where the potential depends onrvia the magnitude
r=

√

rνrν, thusΦ=Φ(r). Then one has

∇μΦ(r)=

dΦ

dr

∇μr.

Use of the definition ofrleads to

∇μr=∇μ

√

rνrν=

1

2

r−^1 ∇μ(rνrν)=r−^1 rν∇μrν=r−^1 rνδμν=r−^1 rμ.

The rule

∇μr=r−^1 rμ=r̂μ (7.13)

is important for many applications. It can be remembered by observing that the direc-

tion of∇μrmust be parallel to theμ-direction sincerdoes not contain any directional

information, and by dimensional considerations, the result of the application of the

nabla operator onrmust be a dimensionless unit vector.

Thus the gradient of a spherical potential function is

∇μΦ(r)=

dΦ

dr

r−^1 rμ=

dΦ

dr

r̂μ. (7.14)

The resulting forceFμ=−∇μΦ(r)is

Fμ=−Φ′r̂μ,Φ′=

dΦ

dr

. (7.15)

Such a force, which is parallel tor, is referred to ascentral force. The attractive

gravitational force between two masses, like the sun and the earth, is of this type.

The same applies to the electrostatic Coulomb force between two charges. Here the

force is repulsive or attractive, when the signs of both charges are equal or opposite,

respectively. The pertaining potential functions, both for gravitation and Coulomb,

are proportional tor−^1. Notice that the interaction between two spherical particles

is described by potential functions which have a dependence onr, in general. For

example, the potential function of two electrically neutral atoms, e.g. Argon atoms,

are of the type(r 0 /r)^12 −(r 0 /r)^6 , wherer 0 is an effective diameter of the atom. The

first term is responsible for the repulsion at short distances, the second one for the

attraction at larger distances.

The torquer×Fvanishes for a central force. This implies that the orbital angular

momentumLis constant. It is recalled thatLis both perpendicular torand to the

velocity. Thus the constant direction ofLimplies that the motion takes place in a

plane. Such a motion is essentially two-dimensional although it takes place in 3D.

The interaction between two particles or extended objects can be described

by a spherical potential function just depending on the inter-particle separation