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Geometrical Optics question

2022 · 25 Jul · Shift 2 · Q69
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Geometrical Optics question

2022 · 25 Jul · Shift 2 · Q69

JEE MainPhysicsGeometrical OpticsNumerical+4 / −1
A convex lens of focal length 20 cm is placed in front of a convex mirror with principal axis coinciding each other. The distance between the lens and mirror is 10 cm. A point object is placed on principal axis at a distance of 60 cm from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is ‾\underline{\hspace{2cm}}​ cm.
Numerical answer
View written solutionFree

Correct answer: 100

  1. Given data
  • Convex lens focal length: fL=+20 cmf_L = +20\,\text{cm}fL​=+20cm
  • Distance between lens and convex mirror: d=10 cmd = 10\,\text{cm}d=10cm
  • Object distance from lens: u1=−60 cmu_1 = -60\,\text{cm}u1​=−60cm

We use the Cartesian sign convention.

  1. First image formed by the lens

For the lens,

1fL=1v1−1u1\frac{1}{f_L} = \frac{1}{v_1} - \frac{1}{u_1}fL​1​=v1​1​−u1​1​

So,

120=1v1−(−160)=1v1+160\frac{1}{20} = \frac{1}{v_1} - \left(-\frac{1}{60}\right) = \frac{1}{v_1} + \frac{1}{60}201​=v1​1​−(−601​)=v1​1​+601​

Thus,

1v1=120−160=260=130\frac{1}{v_1} = \frac{1}{20} - \frac{1}{60} = \frac{2}{60} = \frac{1}{30}v1​1​=201​−601​=602​=301​

Hence,

v1=+30 cmv_1 = +30\,\text{cm}v1​=+30cm

So the lens alone would form an image 30 cm30\,\text{cm}30cm to the right of the lens.

  1. Object for the convex mirror

The mirror is only 10 cm10\,\text{cm}10cm to the right of the lens, while the lens image is at 30 cm30\,\text{cm}30cm from the lens.

Therefore, relative to the mirror, this point is:

30−10=20 cm30 - 10 = 20\,\text{cm}30−10=20cm

to the right of the mirror.

So for the mirror, the object is virtual and lies behind the mirror at distance 20 cm20\,\text{cm}20cm. Hence,

um=+20 cmu_m = +20\,\text{cm}um​=+20cm

(let the mirror image distance be vmv_mvm​).

For mirrors,

1fm=1vm+1um\frac{1}{f_m} = \frac{1}{v_m} + \frac{1}{u_m}fm​1​=vm​1​+um​1​

Thus,

1fm=1vm+120\frac{1}{f_m} = \frac{1}{v_m} + \frac{1}{20}fm​1​=vm​1​+201​
  1. Condition for final image to coincide with the object

The final image after reflection from mirror and refraction again through lens must coincide with the original object, i.e. at 60 cm60\,\text{cm}60cm to the left of lens.

For the second pass through the lens, let the mirror form an image at a point which acts as object for the lens on its right side at distance xxx from the lens.

If this object is at distance xxx to the right of lens, then for the lens on second pass:

u2=+xu_2 = +xu2​=+x

(since object is on the right side of lens for light incident from right). The final image must be at 60 cm60\,\text{cm}60cm left of lens, so

v2=−60 cmv_2 = -60\,\text{cm}v2​=−60cm

Now apply lens formula again:

1fL=1v2−1u2\frac{1}{f_L} = \frac{1}{v_2} - \frac{1}{u_2}fL​1​=v2​1​−u2​1​ 120=1−60−1u2\frac{1}{20} = \frac{1}{-60} - \frac{1}{u_2}201​=−601​−u2​1​ 120+160=−1u2\frac{1}{20} + \frac{1}{60} = -\frac{1}{u_2}201​+601​=−u2​1​ 460=−1u2\frac{4}{60} = -\frac{1}{u_2}604​=−u2​1​ 115=−1u2\frac{1}{15} = -\frac{1}{u_2}151​=−u2​1​

So,

u2=−15 cmu_2 = -15\,\text{cm}u2​=−15cm

This means the object for the second pass must be at 15 cm15\,\text{cm}15cm to the left of the lens.

Therefore the mirror must form its image at a point 15 cm15\,\text{cm}15cm left of the lens. Since the mirror is 10 cm10\,\text{cm}10cm right of the lens, that point is:

10+15=25 cm10 + 15 = 25\,\text{cm}10+15=25cm

to the left of the mirror. So for mirror,

vm=−25 cmv_m = -25\,\text{cm}vm​=−25cm
  1. Use mirror formula

Now apply mirror formula:

1fm=1vm+1um\frac{1}{f_m} = \frac{1}{v_m} + \frac{1}{u_m}fm​1​=vm​1​+um​1​ 1fm=1−25+120\frac{1}{f_m} = \frac{1}{-25} + \frac{1}{20}fm​1​=−251​+201​ 1fm=−4100+5100=1100\frac{1}{f_m} = -\frac{4}{100} + \frac{5}{100} = \frac{1}{100}fm​1​=−1004​+1005​=1001​

Thus,

fm=100 cmf_m = 100\,\text{cm}fm​=100cm

But this is positive, which is consistent with a convex mirror.

  1. Final answer

The focal length of the convex mirror is

100 cm\boxed{100\,\text{cm}}100cm​
  1. Comparison with stored answer

Stored correct answer: 101010

Our derived answer is 100100100, not 101010.

So I do not agree with the stored answer. The likely issue is that the stored answer may contain a typographical error or sign-convention mistake.

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