Lens Equation and Magnification

Magnification

signed height scale

Magnification is the signed ratio between image height and object height.

Example

Magnification is the signed ratio between image height and object height. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Magnification is a signed ratio

Magnification compares image distance to object distance, with a minus sign that records inversion. Here the image distance is 30 metres and the object distance is 15 metres.

m=−vum = -\frac{v}{u}

At fixed object distance, farther images grow

Hold the object distance at 10 metres. A larger image distance gives a larger magnitude of magnification, and the negative sign keeps marking an inverted real image.

uvmhimage10 m10 m−1−4 m10 m20 m−2−8 m10 m30 m−3−12 m\begin{array}{c|c|c|c}u&v&m&h_{\text{image}}\\10\ \text{m}&10\ \text{m}&-1&-4\ \text{m}\\10\ \text{m}&20\ \text{m}&-2&-8\ \text{m}\\10\ \text{m}&30\ \text{m}&-3&-12\ \text{m}\\\end{array}

At fixed image distance, farther objects shrink

Now hold the image distance at 30 metres. Moving the object farther away makes the same image distance a smaller scale factor.

uvmhimage10 m30 m−3−12 m15 m30 m−2−8 m30 m30 m−1−4 m\begin{array}{c|c|c|c}u&v&m&h_{\text{image}}\\10\ \text{m}&30\ \text{m}&-3&-12\ \text{m}\\15\ \text{m}&30\ \text{m}&-2&-8\ \text{m}\\30\ \text{m}&30\ \text{m}&-1&-4\ \text{m}\\\end{array}

Compute the signed scale

The signed magnification is -2. Multiplying the object height by that scale gives signed image height -8 metres, so the image is inverted and taller.

m=−2himage=−8 mm = -2\qquad h_{\text{image}} = -8\ \text{m}
Signed magnificationThe computed negative height draws an inverted image.FFlensobjectimage
optics Negative magnification means the image is inverted; positive magnification means it is upright.