The thin lens equation
Instead of drawing ray diagrams, we can calculate the image distance using the thin lens formula. Here u is the object distance, v is the image distance and f is the focal length, all measured from the optical centre.
Key formula / Key formula
1/f = 1/u + 1/v
Magnification: m = v / u. Lens power: P = 1 / f (in dioptres, D, with f in metres).
Sign convention (real-is-positive)
In the SPM real-is-positive convention: distances of real objects and real images are positive; virtual images are negative. A convex lens has a positive f; a concave lens has a negative f. A negative v tells you the image is virtual and on the same side as the object.
Worked example / Worked example
An object is 15 cm from a convex lens of focal length 10 cm. Find the image distance and magnification.
1/v = 1/f − 1/u = 1/10 − 1/15 = (3 − 2)/30 = 1/30.
v = 30 cm (real, positive).
m = v/u = 30/15 = 2.0 (magnified, inverted).
Worked example / Worked example
An object is 5 cm from a convex lens of focal length 10 cm (magnifying glass). Find the image distance.
1/v = 1/f − 1/u = 1/10 − 1/5 = (1 − 2)/10 = −1/10.
v = −10 cm. The negative sign means the image is virtual and upright, 10 cm on the object's side.
Remember / Remember
- Rearrange carefully: 1/v = 1/f − 1/u.
- A convex lens of f = 0.25 m has power P = 1/0.25 = +4 D.
- A negative image distance means a virtual image.