By Stefan Nanz
Stefan Nanz investigates the need for 3 multipole households in classical electrodynamics. He exhibits that by way of implementing symmetry and parity constraints, it really is adequate to house purely multipole households. this means that the toroidal multipole moments don't symbolize an self sustaining multipole kin, they usually in basic terms emerge within the long-wavelength limit.
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Extra resources for Toroidal Multipole Moments in Classical Electrodynamics: An Analysis of their Emergence and Physical Significance
Hence, the electric mean-square radii can be viewed as a gauge artifact, which can be removed by a diﬀerent choice of A and ϕ. Therefore, it does not matter in the ﬁrst order of the vector potential if the primitive or the traceless multipole moment tensors are used. In higher orders, this is not true. There, it makes a diﬀerence in the ﬁelds if one uses the primitive or the traceless multipole moment tensors [10, p. 28]. We will discuss this in chapter 6. This diﬀerence is caused by the toroidal moments, but also by mean-square radii of the magnetic and toroidal multipole moments.
N) (τ ) = M (−1)n+1 (n + 1)(2n − 1)!! 60) ˆ(n) T n n ˆ˙ (n) . 61) This representation of A now actually contains the physical multipole moment tensors including the toroidal one. Now in every order n, there exist three n-th pole terms. Because of the additional curl of the magnetic multipole moment tensor, they do not appear in the same order when the vector potential is expanded with respect to r /r. The formula for the Tˆ(n) is not in ˆ It would be a closed form, since one needs to use the electric multipole tensors to calculate Λ.
On the other hand, due to numerical or experimental restrictions, the sphere should be as close enough to the sources as possible. Of course, the results for alm and blm have to be independent of the choice of R0 . A suitable choice for R0 therefore usually takes computational or experimental constraints into account . Due to divergence of the radial integral, it is generally not possible to integrate till inﬁnite radius.