Tensors for Physics

(Marcin) #1

164 10 Multipole Potentials


The same applies for a-fold spatial differentiation ofr−^1. In this spirit, Cartesian
tensors of rankare defined by


Xμ 1 μ 2 ···μ≡(− 1 )

∂

∂rμ 1 ∂rμ 2 ···∂rμ

r−^1 =(− 1 )∇μ 1 ∇μ 2 ···∇μr−^1. (10.2)

The tensorial functionsX...approach 0 forr→∞. By definition, these tensors are
symmetric. Whenever two subscripts are equal and summed over, these two spatial
derivatives are equivalent toΔand consequently 0 is obtained. Thus the spatial
differentiation (10.2) yields irreducible tensors of rank. These are thedescending
multipole potentials. Due to the definition (10.2), the-th multipole potential is
related to the−1functionby


Xμ 1 μ 2 ···μ− 1 μ=−


∂rμ

Xμ 1 μ 2 ···μ− 1 =−∇μXμ 1 μ 2 ···μ− 1. (10.3)

10.1.2 Dipole, Quadrupole and Octupole Potentials.


Examples for multipole potential tensors of rank= 1 , 2 ,3arethe


dipole potential


Xμ=r−^3 rμ=r−^2 ̂rμ, (10.4)

thequadrupole potential


Xμν= 3 r−^5

(

rμrν−

1

3

r^2 δμν

)

= 3 r−^5 rμrν = 3 r−^3 ̂rμ̂rν, (10.5)

and theoctupole potential


Xμνλ= 15 r−^7 rμrνrλ = 15 r−^4 ̂rμ̂rν̂rλ. (10.6)

The reason for the namesdipole, quadrupoleandoctupolepotential is seen in
Sect.10.3.


10.1.3 Source Term for the Quadrupole Potential


The formulas presented here and in the following for the multipole potential tensors
are valid forr>0. When a source term is included atr =0, the second rank
multipole tensor obeys the relation

Free download pdf