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Multipole expansion of three collinear charges

For charges

- Q, NQ, - MQ

at

z = d, 0, - d

, respectively,

Φ (r, θ) = \frac{Q}{4 π ϵ _{0}} [\frac{N - M - 1}{r} + \frac{( M - 1 ) d c o s θ}{r ^{2}} - \frac{( M + 1 ) d ^{2} P _{2} ( c o s θ )}{r ^{3}} + O (r^{- 4})] .

The monopole vanishes when

N = M + 1

, the dipole vanishes when

M = 1

, and both vanish when

(N, M) = (2, 1)

.

Table of contents 90 2
- Dipole and quadrupole scaling limits of three collinear charges Multipole expansion of three collinear charges 33
- Transverse force from a collinear electric quadrupole Multipole expansion of three collinear charges 29

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