Checking a lens image against its signs

Use the lens formula 1/f = 1/u + 1/v with the real-is-positive convention, then read the sign of v. Positive v means a real, inverted image. Negative v means a virtual, upright image.

This lesson is part of wave and image interpretation. You can test the same cases in the lens and image-formation explorer.

What are the sign rules?

In the real-is-positive convention, a real object distance u is positive. A real image gives a positive v, and a virtual image gives a negative v. A convex lens has positive f, and a concave lens has negative f.

Decide these signs before you write the formula. Substituting first and deciding the signs later is where errors start.

Worked example 1: object beyond the focal point

A convex lens has f = +10 cm and the object is at u = 15 cm.

1. 1/v = 1/f − 1/u = 1/10 − 1/15 = 3/30 − 2/30 = 1/30.
2. So v = +30 cm. The positive sign means a real, inverted image.
3. Magnification m = v ÷ u = 30 ÷ 15 = 2, so the image is magnified.

The ray diagram agrees: an object between f and 2f gives a real, inverted, magnified image beyond 2f.

Worked example 2: object inside the focal point

Same lens, f = +10 cm, but now u = 6 cm.

1. 1/v = 1/10 − 1/6 = 3/30 − 5/30 = −2/30.
2. So v = −15 cm. The negative sign means a virtual, upright image on the object’s side.
3. m = 15 ÷ 6 = 2.5, so the image is magnified.

This is the magnifying-glass case. The image is larger and upright, and it cannot be projected onto a screen.

Worked example 3: a concave lens

A concave lens has f = −12 cm and the object is at u = 24 cm.

1. 1/v = 1/f − 1/u = −1/12 − 1/24 = −2/24 − 1/24 = −3/24 = −1/8.
2. So v = −8 cm: virtual, upright.
3. m = 8 ÷ 24 = 1/3, so the image is diminished.

A concave lens always gives a virtual, upright, diminished image for a real object, so this result is consistent.

The mistake that costs marks

The usual slip is to drop the negative on a concave lens and use f = +12. Then 1/v = 1/12 − 1/24 = 1/24, so v = +24 cm, and the image looks real and inverted.

That contradicts the ray diagram, which is the warning sign. When the two methods disagree, recheck the sign of f first.

Check yourself

A convex lens has f = +5 cm and an object at u = 20 cm. State v, whether the image is real or virtual, and whether it is magnified or diminished.

What to study next

Go on to explaining a diffraction comparison, or attempt the integrated practice set.

If you want a teacher to check your signs line by line, see online one-to-one Physics tuition. The mistake log helps you record each sign slip.