QUESTION 1: A praying mantis preys along the central axis of a thin symmetrical lens, 20 cm from the lens. The lateral magnification of the mantis provided by the lens is m = 0.25, and the index of refraction of the lens material is 1.65.
(a) Determine the type of image produced by the lens; whether the object (mantis) is inside or outside the focal point; on which side of the lens the image appears; and whether the image is inverted.
(b) What is the magnitude r of the two radii of curvature of the lens?
Solution:
(a) From equation m = – i/p, and the given value for m, we see that
i = mp
= 0.25p
Even without finishing the calculation, we can answer the questions. Because p is positive, i here must be positive. That means we have a real image, which means we have a converging lens (the only lensthat can produce a real image). The object must be outside the focal point (the only way a real image can be produced). And the image is inverted and on the side of the lens opposite the object. (That is how a converging lens makes a real image.)
(b) Because the lens is symmetric, r_{1} (for the surface nearer the object) and r_{2} have the same magnitude r. Because the lens is a converging lens, r_{1 }= +r and r_{2} = r. We have only one question 1/f = (n1) [1/r_{1 }– 1/r_{2}] that includes the radius of curvature of a lens, but we lack a value for f to insert in that equation. We can get f from equation 1/f = 1/p + 1/i if we first find i. So, we must finish the calculation for i, inserting the given value of p and obtaining
i = (0.25) (20 cm)
= 5.0 cm
Now equation 1/f = 1/p + 1/i give us,
1/f = 1/p + 1/i
= [{1/(20 cm)} + {1/(5.0 cm)}]
from which we find, f = 4.0 cm.
Equation 1/f = (n1) [1/r_{1} – 1/r_{2}] gives us,
1/f = (n1) [1/r_{1 }– 1/r_{2}]
= (n1) [1/(+r) – 1/(r)]
or, with known values inserted,
1/(4 cm) = (1.65 – 1) (2/r),
which gives,
r = (2) (0.65) (4.0 cm0
= 5.2 cm
From the above observation we conclude that, the magnitude r of the two radii of
curvature of the lens would be 5.2 cm.
Question 2: A small object is placed 45 cm from a convex refracting surface of radius of curvature 15 cm. If the surface separates air from glass of refractive index 1.5, find the position of the image. Also, determine the first and second principal focal lengths. (IITJEE)
Solution:
Distance of object, u = 45 cm
Radius of curvature, R = +15 cm
μ_{2} = 1.5
μ_{2} = 1
As the object lies in the rarer medium
μ_{2}/v – μ_{1}/u = μ_{2} – μ_{1}/ R
[1.5/v] – [1/(45)] = 0.5/15
= 1/30
1.5/v = 1/30 – 1/45
1.5/v = 1/90
v = 135 cm
Therefore position of the image would be at 135 cm.
First principal focal length is given by
f_{1} =  μ_{1}R/[μ_{2}–μ_{1}]
= – (1×15)/0.5
= 30 cm
Second principal focal length is given by
f_{2} =  μ_{2}R/[μ_{2}–μ_{1}]
= – (1.5×15)/0.5
= 45 cm
From the above observation we conclude that, the first focal length would be at distance 30 cm and second focal length would be at distance 45 cm.
Question 3: An object is placed at a certain distance from a convex lens of focal length 20 cm. Find the distance of the object if the image obtained is magnified 4 times.
Solution:
Case (i): When the image is real,
m = 4
So, v/u = 4 or v = 4u
Now, 1/f = 1/v – 1/u
So, 1/20 = – (1/4u) – (1/u)
= – (1/u) [1+(1/4)]
Or, 1/20 = – (5/4u)
Or, u = (20)× (5/4)
= 25 cm
From the above observation we conclude that, the distance of the object if the image obtained is magnified 4 times would be 25 cm.
Case (ii): When the image is virtual, m = +4
So, v/u = +4 or v = +4u
Again, 1/f = 1/v – 1/u
= (1/4u) – (1/u)
So, 1/20 = (1/u) [(1/4) – 1]
1/20 = (3/4u)
Or, u = (20) (3/4) = 15 cm
From the above observation we conclude that, the distance of the object if the image obtained is magnified 4 times would be 15 cm.
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