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With regards to flux i have always assumed it was defined as $$(1) Something/m^2s$$ For instance $Luminosity/(4\pi r^2)$ Is an example of radiation flux.

Why then is faradys law often cited as the rate of change of magnetic flux is equal in magnitude to the induced emf $$\frac{d\phi}{dt} =\epsilon $$

where $\phi =BA$ So how can we call $\phi$ to be a flux as dimensionally it is not of the same form as (1).

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It depends on it's a flux of what. Usually flux is defined for a vector field. It's obviously so for $\phi(\vec B)$. As to luminosity it too is a vector field: the current density of energy - dimensionally energy per unit area and time. But it also has a direction, characterizing it as a vector.

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