Lens Sign Audit - Light & Optics Step by Step

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\ \text{m}\qquad u = 15\ \text{m}\qquad v = 30\ \text{m}\qquad m = -2

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.

\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}

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.

\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}

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\Rightarrow \text{real},\qquad v<0\Rightarrow \text{virtual}