All light sources should be black, for the following reason. Take the sun, for example. A cone in the retina of the eye collects radiation emanating from a very large number of atoms. These sources are incoherent.
At any given instant there is a near-infinite number of phasors in the complex plane, all of different magnitudes and different phases, rotating at different velocities. Their vector sum is clearly zero.
The same reasoning applies to any object, say a white wall, illuminated with incoherent light. The radiation that reaches a given cone comes from an area that is a large number of wavelengths in diameter. There again, the net field at the cone should be zero, and the wall should appear black.
To explain this paradox, consider $N$ waves of a single frequency and of a given linear polarization but of random amplitudes and phases. The number $N$ is very large. For the $i$-th wave, $E_{i}=E_{m i} \exp j\left(\omega t-\alpha_{i}\right)$ at the cone, and the net $E$ is the sum of the $E_{i}$ 's.
Now the eye is sensitive, not to $E$ but to $\mathscr{S}$, and thus to $E E^{*}$. Show that $\mathscr{S}_{\mathrm{av}}=\sum \mathscr{S}_{i} .$ This means that the net energy flux is equal to the sum of the energy fluxes of the individual waves. ${ }^{\dagger}$