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This Wikipedia article provides a wonderful way to fathom the scale of the tesla. I cannot seem to find a similar set of examples about magnetic flux. I know that the milli-, micro- and nanoweber are the most widely used. Just how strong (weak) is one weber?

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One weber is a unit of magnetic flux – it's $$\int\vec B\cdot d\vec S.$$ So it depends on the area $S$ one integrates over. That's why it's misleading to associate a certain number of webers with the adjectives "strong" and "weak". The magnetic flux isn't really an intensive quantity: it's an extensive quantity of a sort. The larger area you integrate over, the more webers you get.

One tesla is one weber per squared meter. So you may imagine $X$ webers as $X$ teslas integrated over a squared meter and use the table for $X$ teslas from Wikipedia. Of course, you may choose a different area than one squared meter; the numbers must be changed appropriately.

You may get rid of the dependence on the area if you consider the magnetic flux around a magnetic monopole – which hasn't been observed yet. The Dirac quantization considerations tell you that $e\int_D\vec B\cdot d\vec S$ must be a multiple of $2\pi\hbar$ which means that the elementary magnetic monopole has the magnetic flux (over any spherical surface that surrounds it!) $2\pi\hbar/e$, about $10^{-15}$ weber, calculate the exact number if you want.

Note that dimensionally, one weber is one joule per one ampere (and many equivalent ways to write it) so $e$ times the magnetic flux is one ampere-second (coulomb) times one joule over one ampere. The amperes cancel and you're left with joule-seconds, the unit of $\hbar$.

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