Lens Equation and Magnification

Magnification

signed height scale

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

optics Negative magnification means the image is inverted; positive magnification means it is upright.

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