Question

a real inverted image I of an object O is formed by a certain lens (not shown); the object-image separation is d = 31.4 cm, measured along the central axis of the lens. The image is just half the size of the object.​ How far from the object must the lens be placed?​ What is the focal length of the lens?​

Answer 1

To produce this type of image lens Should be convex.

Now given that magnification is 0.5

M = -v/u = -0.5 (negative since image is inverted and real)

Where u = object distance from lens

v = image distance from lens

Now Also given that

d = v + u = 31.4 cm

where -v/u = -0.5

v = 0.5*u

So,

0.5*u + u = 31.4 cm

1.5*u = 31.4 cm

u = 31.4/1.5 = 20.9 cm

Object distance, u = 20.9 cm

Image distance = 0.5*u = 0.5*20.9 = 10.5 cm

Now Using Lens equation

1/f = 1/u + 1/v

f = v*u/(v + u)

f = 20.9*10.5/(20.9 + 10.5)

f = 6.99 cm = focal length of lens

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