Sign and Side Audits

Lens Sign Audit

real and virtual sides

A lens sign audit checks each object and image distance against the sign convention, sorting real and virtual sides so the thin-lens equation 1/f = 1/d_o + 1/d_i closes consistently. The arithmetic is exact only for the stated distances, focal lengths, and side/sign conventions in the ideal thin-lens model. Real measurements add alignment, lens aberrations, finite aperture, thickness, screen/readout calibration, sign-convention bookkeeping, finite precision, and measurement uncertainty.

A converging lens uses the signs of image distance and magnification to record side and orientation. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Start with a checked real image

Use a converging lens with focal length 10 metres and object distance 15 metres. The checked ray construction gives image distance 30 metres and magnification -2.

f=10 mu=15 mv=30 mm=−2f = 10\ \text{m}\qquad u = 15\ \text{m}\qquad v = 30\ \text{m}\qquad m = -2
Lens sign auditThe ray intersection is the middle sign-audit case.FFlensobjectimage

Outside the focus keeps the image real

Hold the focal length at 10 metres and keep the object outside the focus. The image distance stays positive while the negative magnification marks an inverted real image.

fuvm10 m30 m15 m−1210 m20 m20 m−110 m15 m30 m−2\begin{array}{c|c|c|c}f&u&v&m\\10\ \text{m}&30\ \text{m}&15\ \text{m}&\tfrac{-1}{2}\\10\ \text{m}&20\ \text{m}&20\ \text{m}&-1\\10\ \text{m}&15\ \text{m}&30\ \text{m}&-2\\\end{array}
Lens sign auditThe last table row is the checked ray diagram.FFlensobjectimage

Crossing inside the focus changes the side

Now compare two outside-focus rows with one inside-focus row. Inside the focus, the image distance becomes negative and the magnification becomes positive.

fuvm10 m30 m15 m−1210 m15 m30 m−210 m5 m−10 m2\begin{array}{c|c|c|c}f&u&v&m\\10\ \text{m}&30\ \text{m}&15\ \text{m}&\tfrac{-1}{2}\\10\ \text{m}&15\ \text{m}&30\ \text{m}&-2\\10\ \text{m}&5\ \text{m}&-10\ \text{m}&2\\\end{array}
Lens sign auditThe inside-focus row is the checked virtual diagram.FFlensobjectimage

The sign records the image side and orientation

For this lens convention, positive image distance means a real image on the far side. Negative image distance means a virtual image on the object side. The sign of magnification records inverted or upright.

v>0⇒real,v<0⇒virtualv>0\Rightarrow \text{real},\qquad v<0\Rightarrow \text{virtual}
Lens sign auditThe virtual ray diagram and sign table make the same claim.FFlensobjectimage