Page 416 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 32] LENSES 401

In general, an object that is the distance 2 f from a converging lens has a real, inverted image the same size as
the object with an image distance equal to 2 f .

SOLVED PROBLEM 32.10
A key 6 cm long is 100 cm from a converging lens whose focal length is 40 cm. Find the location, size,
and nature of the image.
Here p = 100 cm and f =+40 cm, so the image distance is
pf (100 cm)(40 cm)
q = = = 66.7cm
p − f 100 cm − 40 cm
The image is real since q is positive (Fig. 32.8). The length of the key’s image is
q 66.7cm
h =−h = (−6cm) =−4cm

p 100 cm
The image is inverted (since h is negative) and is smaller than the object.

In general, an object that is farther than 2 f from a converging lens has a real, inverted image smaller than the
object with an image distance between f and 2 f .

SOLVED PROBLEM 32.11

A diverging lens has a focal length of −2 ft. What are the location, size, and nature of the image formed
by the lens when it is used to look at an object 12 ft away?
Here p = 12 ft and f =−2 ft, so the image distance is

pf (12 ft)(−2ft)
q = = =−1.71 ft
p − f 12 ft − (−2ft)
A negative image distance signifies a virtual image. The magnification is

q (−1.71 ft) 1
m =− =− = 0.143 =
p 12 ft 7
The image is erect (since m is positive) and one-seventh the size of the object.

SOLVED PROBLEM 32.12
A double-convex lens has a focal length of 6 cm. (a) How far from an insect 2 mm long should the lens
be held in order to produce an erect image 5 mm long? (b) What is the image distance?

(a) A double-convex lens is always converging, so the focal length of the lens is +6 cm. An erect image means a
positive magnification, which is

h 5mm
m = = = 2.5
h 2mm
Since m =−q/p, the image distance is q =−mp. We proceed by substituting q =−mp in the lens equation
and solving for p:
1 1 1 1 1 1 m − 1 1
= = − = =
p q f p mp f mp f

m − 1 2.5 − 1
p = f = (+6cm) = 3.6cm
m 2.5
(b) q =−mp =−(2.5)(3.6cm) =−9cm
The negative image distance signifies a virtual image (Fig. 32-9).