Mirrors

Concave Mirror Equation

same reciprocal budget

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

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.

1f=1u+1v\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.

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 equationThe middle table row is the checked ray case.Fmirrorobjectimage

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.

fuvm3 m12 m4 m−134 m12 m6 m−126 m12 m12 m−1\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.

1v=16 m−112 m=112 1/m\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 mm=−1himage=−3 mv = 12\ \text{m}\qquad m = -1\qquad h_{\text{image}} = -3\ \text{m}
Concave mirror equationThe equation distance matches the ray intersection.Fmirrorobjectimage
optics Positive image distance for a concave mirror means the real image is in front of the mirror.