Optics

Converging and diverging lenses

A thin lens redirects light by refraction at its two surfaces. For a real object in front of the lens, image position is related to focal length and object distance by the thin-lens equation.

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Definition: Key concepts

Here f is focal length, do object distance, and di signed image distance. A converging lens has f > 0; a diverging lens has f < 0. A real image has di > 0 and can be projected onto a screen; a virtual image has di < 0.

Adjust focal length and object position; follow principal rays, image position, and magnification.

Law and application

Example: Worked example

A converging lens has f=10 cm and an object at do=30 cm. Find the image position and type.

Solution

1/di=1/10−1/30=1/15 cm⁻¹, so di=15 cm. The image is real and inverted; magnification m=−di/do=−0.5, so it is half-size.

Quick check

For a thin lens in air, $1/f=1/d_o+1/d_i$ relates focal length to object and image distances; magnification is $m=-d_i/d_o$ . For a converging lens with $f=10$ cm and an object at $d_o=30$ cm, $d_i=15$ cm and $m=-0.5$ : the image is real, inverted, and half as tall as the object. If the object is inside the focal length, the image is virtual, upright, and magnified. A diverging lens with a real object always forms a virtual, upright, reduced image.Use the same distance unit throughout a calculation. A negative $d_i$ signals a virtual image under the stated convention; check the result with a ray sketch rather than relying on the number alone.

To construct a lens image, a ray parallel to the principal axis passes through the image focus after a converging lens; a ray through the optical centre travels nearly straight. A diverging lens sends a parallel ray outward as if it came from the object focus. For thin lenses in contact in air, optical powers approximately add: $1/f_{eq}=1/f_1+1/f_2$ . This rule explains how camera lens systems combine several elements to correct image defects and adjust focal length.

A converging lens has f=10 cm and an object distance of 30 cm. How far away is the real image?

Where can a real image formed by a converging lens be caught?

References

Young, Freedman (2019). University Physics