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Question

A plano-convex lens P and a concavo-convex lens Q are in contact as shown in figure. The refractive index of the material of the lens P and Q are 1.8 and 1.2 respectively. The radius of curvature of the concave surface of the lens Q is double the radius of curvature of the concave surface. The convex surface of Q is silvered. The radius of curvature of common surface if focal length of the system is 8 cm will be


A
48 cm
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B
24 cm
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C
12 cm
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D
8 cm
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Solution

The correct option is A 48 cm
Given,
refractive index of the material P, μP=1.8
refractive index of the material Q, μQ=1.2
focal length of the system, f=8 cm

Formula for the focal length of the combination of the thin lenses,

1F=1FP+1FQ....(1)

From lens makers formula plano-convex lens P,

1FP=(μP1)(1R11R2)....(2)

where,
R1= Radius of curvature of plane surface=
R2= Radius of curvature of convex surface=2R

Substituting the values in 2,

1FP=(1.81)(112R)

1FP=0.4R.....(3)

Similarly applying the lens makers formula for concavo-convex lens Q,

1FQ=(μQ1)(1R11R2)....(4)

where,
R1= Radius of curvature of convex surface=2R
R2= Radius of curvature of concave surface=R

Substituting the values in 4,

1FQ=(1.21)(12R1R)

1FQ=0.1R.....(5)

Now, from (1), (3) and (5),

1F=0.4R+0.1R=0.5R

Now, combined power of system,
P=PM2PL......(6)

where,
PM= Power of concave mirror=2R
PL= Power of combination of lens=1F=0.5R

Substituting the values in (6),

18=2R2×0.5R

3R=18

R=24 cm

Therefore, radius of curvature of common surface is
2R=2×24=48 cm

Hence, correct option is (a).

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