Why does the sum of incoherent light sources result in non-zero intensity?

[fantispug]

I was thinking about a collection of monochromatic light sources with a randomly oriented phase - for example in a sodium lamp the atoms will radiate almost independently, so they should act as such a collection of incoherent light sources.

Assume for simplicity each light source is classical and has the same electric field magnitude $$E_0$$. The electric field of each of them at a point in space would be $$E_0e^{(i\varphi)}$$ (for some $$\varphi$$ dependent on the individual light source and the position in space) and the total electric field would be the sum over all of these electric fields at that point.

Given that the phases of the sources are unrelated it should be expected that any value of $$\varphi$$ could occur with equal probability, that is the phasor (complex vector) corresponding to the electric field could be oriented in any direction with equal probability. The sum over a large number of these sources should thus give something close to zero.

However this would imply zero intensity and since we can observe the light from a sodium lamp this doesn't make sense.
What am I doing wrong here?

 

[cesiumfrog]

Isn't the intensity related to the average square of the amplitude, rather than the average amplitude?

 

[fantispug]

Yeah, you're completely right. I don't know why I was thinking the square of the average, rather than the average of the square (since the electric field at a point can tell you nothing about the power passing through that point). So the cross-terms from the square - the interference terms - will average to zero, and the intensity will just be the sum of the individual intensities!
Thank you so much, that makes much more sense

 

[cesiumfrog]

Pleased to help