Tensors for Physics

82 7 Fields, Spatial Differential Operators

function. Some examples for the special forms of potential functions discussed in
Sect.7.1, are presented next.

(i) Planar Geometry

Let the potential depend onrviax=e·r=eνrν, whereeis a constant unit vector.
Then the chain rule, with the help of (7.9), leads to,

∇μΦ(x)=

dΦ

dx

∇μeνrν=

dΦ

dx

eμ. (7.11)

Clearly, one hasFμ∼eμ, the direction of the force is constant. In this very special
case, where the potential is linear inx, also the force strength is constant. Such a
force field is referred to as ahomogeneous field. The gravitational field above a flat
surface on earth is of this type, as long as the height over ground, corresponding to the
variablex, is very small compared with the diameter of the earth. Another example
of an approximately homogeneous field is the electric field between the charged flat
plates of an electric capacitor.

(ii) Cylindrical Geometry

LetΦbe a function of the distanceρ=

√

rν⊥rν⊥of pointrfrom an axis parallel the
the constant unit vectore.Herer⊥ν =rν−eνeλrλis the projection ofronto a plane
perpendicular toe. Use of the chain rule yields

∇μΦ(ρ)=

dΦ

dρ

∇μρ.

Furthermore, one obtains

∇μρ=∇μ

√

rν⊥r⊥ν=

1

2

ρ−^1 ∇μ

(

rν⊥rν⊥

)

=ρ−^1 rν⊥∇μrν⊥=ρ−^1 rν⊥(δμν−eμeν)=ρ−^1 r⊥μ.

Thus the gradient of the potential function is found to be

∇μΦ(ρ)=

dΦ

dρ

ρ−^1 rμ⊥=

dΦ

dρ

̂r⊥

μ, (7.12)

wherer̂μ⊥is the radial unit vector pointing outward from the cylinder axis.

Notice: both the planar and the cylindrical geometry as considered here have
cylindrical symmetry since there is a preferential direction parallel to the constant
vectore. In both cases, the torqueTμ=εμνλrνFλ=−εμνλrν∇λΦis proportional
toεμνλrνeλ. Thus only the component of the orbital angular momentumLwhich is
parallel toeis conserved. The other two components ofL, which are perpendicular
to the preferential direction, do change in time.