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From Eq. 9 and geometric considerations it also follows that the electric multipoles of order N+1 are identically zero and, for even N, also all the odd electric multipoles vanish.
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From Eq. 9 and geometric considerations it also follows that the electric multipoles of order N+1 are identically zero and, for even N, also all the odd electric multipoles vanish.
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49649100325
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It may also be proven that the even magnetic multipoles are all identically zero when N is even.
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It may also be proven that the even magnetic multipoles are all identically zero when N is even.
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39
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49649090976
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For the case N=1, the electric-dipole contribution is evidently not canceled in the summation, consistent with the fact that we are dealing with just an isolated polarizable particle with its own electric-dipole moment.
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For the case N=1, the electric-dipole contribution is evidently not canceled in the summation, consistent with the fact that we are dealing with just an isolated polarizable particle with its own electric-dipole moment.
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40
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49649124757
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As an aside, it may be noticed that by substituting expressions 1 12 in Eq. 16, we derive the extracted power from the nanoloop to be equal to Pext = (N/2) Re [iω p Eimp], i.e., the sum of the powers extracted by each of the nanoparticles from the impressed electric field, which validates the previous analysis.
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As an aside, it may be noticed that by substituting expressions 1 12 in Eq. 16, we derive the extracted power from the nanoloop to be equal to Pext = (N/2) Re [iω p Eimp], i.e., the sum of the powers extracted by each of the nanoparticles from the impressed electric field, which validates the previous analysis.
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41
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It is worth noting that the first-order Taylor series of Eq. 20 is consistent with Eq. 1 and that the excitation proposed in Ref. to isolate the magnetic response of the nanoloop, composed of Npw plane waves symmetrically launched and summing in phase their magnetic fields at the center of the loop, coincides with Eq. 20 when Npw →.
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It is worth noting that the first-order Taylor series of Eq. 20 is consistent with Eq. 1 and that the excitation proposed in Ref. to isolate the magnetic response of the nanoloop, composed of Npw plane waves symmetrically launched and summing in phase their magnetic fields at the center of the loop, coincides with Eq. 20 when Npw →.
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43
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49649083441
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In the formula ζ is the Riemann zeta function. The upper limit to this inequality is obtained taking the limit for N→ of the summation in Eq. 26. The sum is a monotonic function of N rapidly converging to its horizontal asymptote.
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In the formula ζ is the Riemann zeta function. The upper limit to this inequality is obtained taking the limit for N→ of the summation in Eq. 26. The sum is a monotonic function of N rapidly converging to its horizontal asymptote.
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