Physics · Ch 6 — Optics
The Mirror Equation
The Mirror Equation
The mirror equation relates object distance , image distance , and focal length for any spherical mirror: . It is derived for an object placed beyond the centre of curvature of a concave mirror, using three paraxial rays traced from the tip : one parallel to the axis (reflecting through the focus ), one through the pole (reflecting symmetrically), and one through the centre of curvature (retracing its own path); these three reflected rays meet at , locating the real, inverted image . The similar triangles and this construction creates, combined with the Cartesian sign convention applied to every distance, simplify step by step down to the mirror equation. Although derived for this one speci …
What this figure shows. An object AB stands beyond the centre of curvature of a concave mirror. Three paraxial rays are traced from its tip B: one parallel to the axis, striking near the pole at D and reflecting through the focus F; one through the pole P itself, reflecting symmetrically; and one through the centre of curvature C, striking the mirror at E and retracing its own path. All three reflected rays meet at B', locating the real, inverted image A'B'. The similar triangles this construction creates (BPA with B'PA', and DPF with B'A'F) are what the mirror equation 1/v + 1/u = 1/f is algebraically extracted …
Worked out. An object sits 20.0 cm from a concave mirror of focal length 15.0 cm, so by the sign convention u = -20 cm and f = -15 cm. Substituting into the mirror equation 1/v = 1/f - 1/u gives 1/v = -1/15 - (-1/20) = -1/60, so v = -60.0 cm: the image forms 60.0 cm to the left of the mirror (in front of it, on the same side as the object), meaning a screen placed there would catch a genuinely real image, which is why the answer to where the screen should go is exactly 60.0 cm from the mirror. The magnification is m = -v/u = -(-60)/(-20) = -3, and because this value is negative the image is inverted (upside down relative to the objec …