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Transverse force from a collinear electric quadrupole

For charges

- Q, 2 Q, - Q

at

z = d, 0, - d

, the exact force on a charge

- Q

at

(x, 0, 0)

is

F = - \frac{Q ^{2} x}{2 π ϵ _{0}} (\frac{1}{∣ x ∣ ^{3}} - \frac{1}{( x ^{2} + d ^{2} ) ^{3/2}}) x .

It points toward the origin for every nonzero

x

.

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Solution