The text produced a spherical multipole expansion for the magnetic scalar potential (r) based on the identity A Cartesian expansion for the scalar potential can

The text produced a spherical multipole expansion for the magnetic scalar potential ψ(r) based on the identity

A Cartesian expansion for the scalar potential can be developed from the same starting point.

(a) Expand |r − r'|⁻¹ and confirm that

(b) Show that the dipole moment can be written in the familiar form

(c) Show that the magnetic quadrupole can be written as

(d) Prove that m ⁽²⁾ _ij in part (c) is traceless.

(e) Rewrite the formula in part (c) as

(f) Prove that the magnetic quadrupole moment M⁽²⁾ = d³ r (r × j)r produces the same quadrupole scalar potential as m⁽²⁾ in part (c). M⁽²⁾ is the moment which appears in the Cartesian multipole expansion for the vector potential derived in the text.

-r- = I= = r po 4 r. V' xj (r') |rr| dr'.