Question

Question on Polariser

Answer 1

The red part

Answer 2

why?

Answer 3

in math terms, I have no problem with calculus

Answer 4

polarizers are crystals which have an internal optical axis. They allow Electromagnetic Oscillations parallel to their optical axis and forbid oscillations perpendicular to their axis.
Also note that Intensity of light is proportional to the square of the amplitude of the Electric Vector. So if our oscillation is E0Cos(wt) then
\[I \alpha E _{0}^{2}\]
In your case,
first Unpolarized light enters polarizer 1, so only the Electric field vectors along the optical axis are allowed and hence the rest of the light is cut off.
Therefore from polarizer 1, the light that comes out has a reduced Intensity \[I _{1}\]
Now your second polarizer is aligned such that its optical axis is at an angle to the first polarizer. Let this angle be theta.
Then only the component of the Electric field vector along the second crystal's optical axis are allowed to be transmitted. This component is
\[E_{2} = E _{0}*\cos (\theta)\]
Hence, the corresponding Intensity will be
\[I _{2} \alpha E _{2}^{2}\]