Concave Mirror Equation - Optics Step by Step

The spherical-mirror equation uses the same reciprocal distance budget as a thin lens, with mirror sign conventions.

optics Positive image distance for a concave mirror means the real image is in front of the mirror.

Example

The spherical-mirror equation uses the same reciprocal distance budget as a thin lens, with mirror sign conventions. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Use the same distance budget

The spherical-mirror model uses the same reciprocal distance budget as the thin-lens model, with mirror sign conventions.

\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

Fixed concave focus scans object distance

Hold the concave focal length at 6 metres. Moving the object closer to the focus pushes the real image farther away and increases the inverted scale.

\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}

Fixed object distance scans focal length

Now hold the object distance at 12 metres. A longer concave focal length makes the image distance stretch farther from the mirror.

\begin{array}{c|c|c|c}f&u&v&m\\3\ \text{m}&12\ \text{m}&4\ \text{m}&\tfrac{-1}{3}\\4\ \text{m}&12\ \text{m}&6\ \text{m}&\tfrac{-1}{2}\\6\ \text{m}&12\ \text{m}&12\ \text{m}&-1\\\end{array}

Subtract the object reciprocal

For this concave case, the image reciprocal is positive.

\frac{1}{v} = \frac{1}{6\ \text{m}} - \frac{1}{12\ \text{m}} = \tfrac{1}{12}\ 1/\text{m}

Invert to get the image distance

The image distance is the same 12 metres in front as the object, so the image sits at the object's location. The signed image height is -3 metres, so the real image is the same size but inverted.

v = 12\ \text{m}\qquad m = -1\qquad h_{\text{image}} = -3\ \text{m}