• Object distance p is positive for real objects located to the left of the lens and negative for virtual objects located to the right of the lens.
• Image distance q is positive for real images formed to the right of the lens and negative for virtual images formed to the left of the lens.
• The focal length f is positive for a converging lens, negative for a diverging lens.
• The radius of curvature r is positive for a convex surface, negative for a concave surface.
• Transverse distances (ho and hi) are positive above the optical axis, negative below.

Now let’s apply Equations 3-10, 3-11, and 3-12 in several examples, where the use of the sign convention is illustrated and where the size, orientation, and location of a final image are determined.

Example 8

A double-convex thin lens such as that shown in Figure 3-21 can be used as a simple “magnifier.”

(a) What is its focal length in air?
(b) What is its focal length in water (n = 1.33)?
(c) Does it matter which lens face is turned toward the light?
(d) How far would you hold an index card from this lens to form a sharp image of the sun on the card?

Solution:

(a) Use the lensmaker’s equation. With the sign convention given, we have ng = 1.52, n = 1.00, r1 = +20 cm, and r2 = − 15 cm. Then

So f = +16.5 cm (a converging lens, so the sign is positive, as it should be)

f = 60 cm (converging but less so than in air)

(c) No, the magnifying lens behaves the same, having the same focal length, no matter which surface faces the light. You can prove this by reversing the lens and repeating the calculation with Equation 3-11. Results are the same. But note carefully, reversing a thick lens changes its effect on the light passing through it. The two orientations are not equivalent.