On an optics bench, an object and a viewing screen are fixed 75cm apart. A concave spherical mirror is mounted on a carriage that can slide along the bench between the object and the screen. When the mirror is moved along the bench, a sharp real image of the object is observed on the screen at two distinct mirror positions separated by 20cm.
(1) Derive an algebraic expression that relates the focal length of the mirror to the fixed object–screen distance L and the separation d between the two mirror positions that form sharp images.
(2) Use the given values L=75cm and d=20cm to calculate the numerical value of the mirror’s focal length, stating the appropriate sign.
(3) For the mirror position closer to the object, determine the magnification of the image formed on the screen.
(4) Explain why two distinct mirror positions exist only when the fixed distance L exceeds a certain minimum value. Derive that minimum distance in terms of the mirror’s focal length and interpret its physical meaning.