A metal plate is manufactured so that its surface roughness increases linearly from one edge to the other over a horizontal width of 20cm. The plate is oriented vertically and is illuminated head-on by a uniform, collimated white beam of power density I0=200Wm−2.
At the left edge (x=0) the surface behaves as an ideal mirror that sends 90% of the incident power into the exact specular direction. At the right edge (x=20cm) the surface behaves as a perfectly diffuse reflector that sends 90% of the incident power uniformly into the forward hemisphere. Between these edges the fraction of incident power sent into the specular direction decreases linearly with x.
A white projection screen is placed 1.0m in front of the plate, parallel to it. Ignore absorption and assume that the 10% of power not reflected is lost inside the plate.
Address the following tasks.
(a) For a point located at position x along the plate (measured from the left edge), write an expression for the fraction fspec(x) of incident power that reaches the screen at the specular point corresponding to that x.
(b) For the same point x, state the fraction of incident power that is scattered diffusely toward the screen. Justify your expression briefly.
(c) Derive a functional form for the power per unit width Ps(x) arriving at the screen from specular reflection along the plate. Assume the incident beam fully covers the plate and remains collimated over the short 1.0m distance.
(d) Sketch on a single set of axes the qualitative variation of Ps(x) across the screen width and the corresponding diffuse contribution per unit width. Clearly label key features at x=0 and x=20cm and discuss which region of the screen will appear brightest overall.