Displacement Method: Conjugate Points

The displacement method is a practical laboratory technique for measuring the focal length of a convex lens accurately, without needing to know either the object distance or the image distance separately or precisely. An object and a screen are fixed a distance $D$ apart on an optical bench, with $D$ kept greater than four times the (roughly estimated) focal length of the lens. A convex lens, mounted on a holder that can slide along the bench between the object and the screen, is then found to produce a sharp, focused image on the screen at two distinct positions as it is slid along -- once relatively close to the object (giving a large, magnified real image on the screen) and once relatively close to the screen (giving a small, diminished real image on the screen). These two lens positions, at which object and image distances are simply interchanged (the object distance in the first position equals the image distance in the second, and vice versa), are called conjugate points, and this reversibility itself follows directly from the fact that the thin lens formula treats $u$ and $v$ symmetrically. If $d$ is the measured distance between these two lens positions, then simple algebra applied to the two lens-formula equations for the two positions (using $u_1=-\frac{D-d}{2}$ and $u_1=-\frac{D+d}{2}$ for the two positions, together with $v_1+u_1$-type relations fixed by $D$) yields the elegant result $f=\frac{D^2-d^2}{4D},$ expressing the focal length purely in terms of the two directly-m …