The lens formula

The idea

The lens formula 1/v − 1/u = 1/f relates object distance u, image distance v and focal length f, and is valid in all situations for any spherical lens, using the sign convention.

Find the image without drawing

Mirrors have a formula that links u, v and f. Lenses have one too, and it is a little different.

Which quantities does the lens formula connect?

• The object distance, the image distance and the focal length
• The object height and the colour of the light
• Only the thickness of the glass

Check the formula

Move the object. Compare 1/v − 1/u with 1/f. Do this for the convex lens and then for the concave one.

They match

For both lenses, at every distance, 1/v − 1/u gave the same number as 1/f. Note the minus sign between 1/v and 1/u. For a lens it is different from the mirror formula.

The lens formula is 1/v − 1/u = 1/f. It relates the object distance u, the image distance v and the focal length f. It is valid in all situations for any spherical lens, if the signs are used with care.

Notes

1/v − 1/u = 1/f is valid in all situations for any spherical lens. Use the signs with care.

Check yourself

Which is the lens formula?

Example 9.4 of the book: a convex lens has f = +10 cm and the object is at u = −15 cm. Use 1/v − 1/u = 1/f to find v, in cm.

Answer: 30 cm

1/v = 1/u + 1/f = −1/15 + 1/10 = 1/30, so v = +30 cm, on the other side of the lens.

Example 9.3 of the book: a concave lens has f = −15 cm and the object is at u = −30 cm. Find v, in cm.

Answer: -10 cm

1/v = 1/u + 1/f = −1/30 − 1/15 = −3/30, so v = −10 cm, on the same side as the object.

For which lenses and object positions is the lens formula valid?

• 1/v + 1/u = 1/f. That is the mirror formula. For a lens there is a minus sign between 1/v and 1/u.
• 1/v − 1/u = f. On the right side we need 1/f, the reciprocal of the focal length, and not f itself.
• 1/v − 1/u = 1/f — correct. Yes!
• In all situations, for any spherical lens — correct. Yes!
• Only for convex lenses. The formula is general: it is valid for any spherical lens, convex or concave.
• Only when the image is real. The formula is valid in all situations, for real and for virtual images.