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Geometrical Optics question

2016 · 10 Apr · Shift 1 · Q66
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Geometrical Optics question

2016 · 10 Apr · Shift 1 · Q66

JEE MainPhysicsGeometrical OpticsMCQ+4 / −1
A hemispherical glass body of radius 10 cm and refractive index 1.5 is silvered on its curved surface. A small air bubble is 6 cm below the flat surface inside it along the axis. The position of the image of the air bubble made by the mirror is seen : JEE Main 2016 (Online) 10th April Morning Slot Physics - Geometrical Optics Question 199 English
  1. A
    14 cm below flat surface
  2. B
    30 cm below flat surface
  3. C
    20 cm below flat surface
  4. D
    16 cm below flat surface
View written solutionFree

Correct answer: C

  1. Interpret the geometry
  • Radius of hemispherical glass body: R=10 cmR = 10\,\text{cm}R=10cm
  • Refractive index of glass: μ=1.5\mu = 1.5μ=1.5
  • The curved surface is silvered, so it acts as a spherical mirror.
  • The air bubble lies on the axis, 6 cm6\,\text{cm}6cm below the flat surface.

Since the center of the hemisphere is at the flat face center, the pole of the curved mirror is 10 cm10\,\text{cm}10cm below the flat surface.

So the bubble is at a distance from the curved mirror pole: u=10−6=4 cmu = 10 - 6 = 4\,\text{cm}u=10−6=4cm

  1. Image formed by the silvered curved surface (mirror action inside glass)

The silvered curved surface behaves like a concave mirror for rays inside the glass.

For a spherical mirror, f=R2=102=5 cmf = \frac{R}{2} = \frac{10}{2} = 5\,\text{cm}f=2R​=210​=5cm

Using mirror formula, 1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v}f1​=u1​+v1​

Substitute u=4 cmu=4\,\text{cm}u=4cm and f=5 cmf=5\,\text{cm}f=5cm: 15=14+1v\frac{1}{5} = \frac{1}{4} + \frac{1}{v}51​=41​+v1​ 1v=15−14=4−520=−120\frac{1}{v} = \frac{1}{5} - \frac{1}{4} = \frac{4-5}{20} = -\frac{1}{20}v1​=51​−41​=204−5​=−201​ v=−20 cmv = -20\,\text{cm}v=−20cm

The negative sign means the image lies behind the mirror (with this sign convention), i.e. on the other side of the pole by 20 cm20\,\text{cm}20cm.

Since the mirror pole is 10 cm10\,\text{cm}10cm below the flat surface, the image position relative to the flat surface is: 10+20=30 cm10 + 20 = 30\,\text{cm}10+20=30cm

This is the image formed by the mirror inside glass.

  1. Seen through the plane flat surface

Now this image at 30 cm30\,\text{cm}30cm below the flat surface is viewed from air through the plane glass surface.

For refraction at a plane surface, μ2v−μ1u=0\frac{\mu_2}{v} - \frac{\mu_1}{u} = 0vμ2​​−uμ1​​=0 which gives apparent depth relation: apparent depth=real depthμ\text{apparent depth} = \frac{\text{real depth}}{\mu}apparent depth=μreal depth​

Here real depth in glass =30 cm= 30\,\text{cm}=30cm, and μ=1.5\mu = 1.5μ=1.5.

Hence apparent depth from air is dapp=301.5=20 cmd_{\text{app}} = \frac{30}{1.5} = 20\,\text{cm}dapp​=1.530​=20cm

  1. Final answer

Thus, the image of the air bubble made by the mirror is seen at 20 cm below the flat surface\boxed{20\,\text{cm below the flat surface}}20cm below the flat surface​

So the correct option is C.

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