Sign and Side Audits

Mirror Sign Audit

concave and convex rows

Concave and convex mirrors use the same reciprocal budget, but the focal-length sign changes where the image appears. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Concave mirrors can make real front images

Keep the concave focal length positive and move the object. The positive image distances are front-side real images, and the negative magnifications mark inversion.

fuvm6 m18 m9 m−126 m12 m12 m−16 m9 m18 m−2\begin{array}{c|c|c|c}f&u&v&m\\6\ \text{m}&18\ \text{m}&9\ \text{m}&\tfrac{-1}{2}\\6\ \text{m}&12\ \text{m}&12\ \text{m}&-1\\6\ \text{m}&9\ \text{m}&18\ \text{m}&-2\\\end{array}
Concave mirror sign auditThe middle row is the checked reflected-ray case.Fmirrorobjectimage

Changing mirror type changes the sign pattern

Hold the object distance fixed and compare a concave row with two convex rows. Convex focal lengths are negative, and the image distance stays negative in these checked rows.

fuvm6 m12 m12 m−1−6 m12 m−4 m13−3 m12 m−125 m15\begin{array}{c|c|c|c}f&u&v&m\\6\ \text{m}&12\ \text{m}&12\ \text{m}&-1\\-6\ \text{m}&12\ \text{m}&-4\ \text{m}&\tfrac{1}{3}\\-3\ \text{m}&12\ \text{m}&\tfrac{-12}{5}\ \text{m}&\tfrac{1}{5}\\\end{array}
Convex mirror sign auditThe middle table row is the checked convex case.Fmirrorobjectimage

Convex mirrors keep upright virtual images

Hold the convex focal length fixed and move the object. The image distance remains negative, while positive magnification keeps the image upright.

fuvm−6 m6 m−3 m12−6 m12 m−4 m13−6 m18 m−92 m14\begin{array}{c|c|c|c}f&u&v&m\\-6\ \text{m}&6\ \text{m}&-3\ \text{m}&\tfrac{1}{2}\\-6\ \text{m}&12\ \text{m}&-4\ \text{m}&\tfrac{1}{3}\\-6\ \text{m}&18\ \text{m}&\tfrac{-9}{2}\ \text{m}&\tfrac{1}{4}\\\end{array}
Convex mirror sign auditThe middle table row is the checked virtual image.Fmirrorobjectimage

Mirror signs separate front and behind

For the mirror convention used here, positive image distance is a front-side real image. Negative image distance is a virtual image behind the mirror.

v>0⇒front real,v<0⇒behind virtualv>0\Rightarrow \text{front real},\qquad v<0\Rightarrow \text{behind virtual}
Convex mirror sign auditThe sign rule is grounded in the checked convex row.Fmirrorobjectimage