Lens Equation and Magnification

Lens Equation

distance budgets for rays

The thin-lens equation connects focal length, object distance, and image distance.

Example

The thin-lens equation connects focal length, object distance, and image distance. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Three distances are linked

A thin lens connects three distances: focal length, object distance, and image distance. If you know two of them, the lens equation determines the third.

1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v}
Three linked distancesThe ray diagram supplies the same f, u, and v distances used in the equation.FFlensobjectimage

Fixed focal length scans object distance

Hold the focal length at 10 metres. Moving the object changes the image distance and can even move the image to the object side when the object is inside the focus.

fuvm10 m20 m20 m−110 m15 m30 m−210 m5 m−10 m2\begin{array}{c|c|c|c}f&u&v&m\\10\ \text{m}&20\ \text{m}&20\ \text{m}&-1\\10\ \text{m}&15\ \text{m}&30\ \text{m}&-2\\10\ \text{m}&5\ \text{m}&-10\ \text{m}&2\\\end{array}
Object-distance scanThe middle row is the checked ray case; the table shows nearby exact placements.FFlensobjectimage

Fixed object distance scans focal length

Now hold the object distance at 12 metres. A longer focal length makes the image land farther away, so the same equation row shows the geometry stretching.

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}
Focal-length scanThe middle table row is the checked ray case for this fixed-object-distance scan.FFlensobjectimage

Solve for the image distance

To find the image distance, move the object-distance part to the other side. The reciprocals are the pieces that add.

1v=1f−1u\frac{1}{v} = \frac{1}{f} - \frac{1}{u}
Solved image distanceThe solved distance is the point where the checked rays intersect.FFlensobjectimage
optics The equation is an ideal thin-lens model. The diagrams still come from checked ray geometry.