Find an expression for intensity of transmitted light when a polaroid sheet is rotated between two crossed polaroids. In which position of the polaroid sheet will the transmitted intensity be maximum?

Let I0 be the intensity of unpolarised light, incident on the first polaroid. The intensity of light transmitted from the first polaroid will be, I1 = I0cos²θ, or I1 = I0 × 1/2 = I0/2 eq1. Let θ be the angle between the transmission axes of the first and second polaroid (which is placed between two crossed polaroids) and φ be the angle between the second and third polaroid, then θ + φ = 90 (as first and third polaroids are perpendicular to each other), or φ = 90 - θ. Now, the intensity of light transmitted from the second polaroid, I2 = I1cos²θ = (I0/2)cos²θ, and the intensity of light transmitted from the third polaroid, I3 = I2cos²φ = (I0/2)cos²θ × cos²φ, I3 = (I0/2)cos²θ × cos²(90 - θ), or I3 = (I0/2)cos²θsin²θ, or I3 = (I0/2)cos²θsin²θ, or I3 = (I0/2)(sin22θ/4). This is the required expression. Now I3 will be maximum when sin2θ is maximum i.e. sin2θ = 1 = sin90, or 2θ = 90, or θ = 45°