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Here's the magnetic field produced by a quadrupole

enter image description hereCiting from Barr, Giles; Devenish, Robin; Walczak, Roman; Weidberg, Tony Particle Physics in the LHC Era (Oxford Master Series in Physics) (p.55,56). OUP Oxford:

One says that the quadrupole is focusing in one plane and defocusing in another plane, perpendicular to the first. There is a complete analogy with geometrical optics, and for a quadrupole of length $l$, with quadrupole strength $k$, one can define its focal length $f$: $$ \frac 1 f = kl \qquad $$

Where does this formula come from?

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The power of a lens is 1/f. Also sometimes called the strength of the lens. Stronger lenses bend the light more. The focus closer to the geometric optics lens. So I think that he is just saying a longer quadrupole of a given strength and the product of L and k is the power of the quadrupole lens and the focal point is closer to the lens if the product is higher.

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