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We were discussing Kirchoff's theorem on blackbody radiation in class today and the equation was: $$e_{f}= J(f,T)$$

$e_f$ was defined as power per unit area per unit frequency. What does per unit frequency refer to here? I know what per unit time means but don't understand 'per unit frequency'.

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The blackbody radiation is radiation with not only one frequency (e.g. like in a laser beam). Instead it is a mixture of many frequencies (essentially all frequencies from $0$ to $\infty$).

Therefore it doesn't make much sense to ask for the radiation power at a single frequency $f$. But it makes perfect sense to ask for the radiation power in a frequency range (from $f_1$ to $f_2$).

Power per unit area and per unit frequency just means the power emitted from an area of $1$ m$^2$ in a frequency range of $1$ Hz (for example in the range from $500$ THz to $500.000000000001$ THz).

The spectrum below shows the radiation power per area (measured in m$^2$) and per frequency range (measured in Hz) as it varies depending on the frequency (here given in THz).

enter image description here
image from Energy eduation | Blackbody radiation

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To begin with, power per unit area gives the intensity. So, basically, $e_f$ is denoting the Intensity per unit frequency. It can give an idea about intensity of radiation absorbed by the black body for a range of frequency.

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