A student is reading a book located near the edge of a circular table of radius R . A point source of light is suspended directly above the centre of the table. Determine the height of the source above the table required to yield maximum illuminance at the book's position.

Step-by-step solution

Let the height of the source above the center of the table be h . The distance from the source to the book at the edge of the table is r = R^2 + h^2 . The illuminance E at the book's position is given by E = I r^2 , where I is the luminous intensity of the source and is the angle of incidence. From the geometry, = h r = h R^2 + h^2 . Substituting the values of r and , we get: E = I h (R^2 + h^2)^ 3/2 For maximum illuminance, dE dh = 0 . dE dh = I [ 1 (R^2 + h^2)^ 3/2 - h 3 2 (R^2 + h^2)^ 1/2 2h (R^2 + h^2)^3 ] = 0