One of my school exam questions asked to show(mathematically of physically) that the intensity of unpolarised light passing through a polarizer is halved. I found this rather difficult to prove, but i made an attempt anyway.
MY ATTEMPT: Malus law: if we have 2 polarizers P1 and P2 inclined at angle x with each other, the intensity passing through P2 is $I_o$$cos^2(x)$ where $I_o$ is the intensity of light after P1. I cannot use this formula directly in this case because of the definition. But, if i understand correctly(which is not likely) P1 simply makes the unpolarised light plane polarised. So suppose now we consider a beam of unpolarised light. This can be thought of as a mixture of many plane polarised lights.consider any one of them and apply malus law. Now $I_o$ is the original intensity and x is the angle of the pass axis w.r.t the chosen plane polarised light. Since this can be chosen absolutely randomly, angle x can vary randomly from 0 to 2pi. We can theorefore find the EXPECTED transmitted intensity to be the average of $cos^2(x)$ as x varies from 0 to 2pi, which returns 1/2. Is this a correct proof of the question? I am not satisfied myself. Any help with regard to my proof woulf be appreciated. Also, i would like a physical explanation if possible.