Tensors for Physics

168 10 Multipole Potentials

cf. (8.81). This force can be written as the negative gradient ofqφ,viz.Fμ=−∇μqφ,
where

φ=

Q

4 πε 0

1

r

,

is the electrostatic potential. ForNchargesqi, located at positionsrithe correspond-
ing expression is

φ=

1

4 πε 0

∑N

i= 1

qi

|r−ri|

. (10.21)

The generalization to a continuous charge densityρ(r′)is the electrostatic potential
φ(r)given by the integral

φ(r)=

1

4 πε 0

∫

ρ(r′)

|r−r′|

d^3 r′. (10.22)

By analogy to the mass density, cf. Sect.8.3.2, the charge density can also be written
asρ(r′)=

∑

iqiδ(r

′−ri). Insertion of this expression for the charge density into

(10.22) yields (10.21).

10.3.2 Expansion of the Electrostatic Potential

In the following, it is understood that the charge distribution is centered aroundr′= 0
and thatris a point further away from this center than any of the charges generating
the electrostatic potential, cf. Fig.10.1.
So it makes sense to expand|r−r′|−^1 , occurring in (10.22), aroundr′=0.
Due to

∂

∂rμ′

|r−r′|−^1 =−

∂

∂rμ

|r−r′|−^1 ,

the Taylor series expansion of|r−r′|−^1 reads

Fig. 10.1Charge cloud
centered aroundr′=0. The
pointris outside of the cloud