An object is placed at a distance of 10 cm from a co-axial combination of two lenses A and B in contact. The combination forms a real image three times the size of the object. If lens B is concave with a focal length of 30 cm, what is the nature and focal length of lens A ?a)Concave, 12 cmb)Convex, 18 cmc)Convex, 12 cmd)Convex, 6 cmCorrect answer is option 'D'. Can you explain this answer?

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Answer

Given:
- Distance of the object from the co-axial combination of lenses A and B = 10 cm
- The combination forms a real image three times the size of the object
- Lens B is concave with a focal length of 30 cm

To find:
- Nature and focal length of lens A

Solution:

Step 1: Identify the type of image formed
Since the image formed is real and larger than the object, it indicates that the lenses combination acts as a magnifying glass or a magnifier. This means that the image is virtual, erect, and magnified.

Step 2: Determine the relationship between the image distance and object distance
Using the magnification formula:
magnification (m) = height of the image (h') / height of the object (h)
Given: h' = 3h (image is three times the size of the object)
Therefore, magnification (m) = 3

Using the lens formula:
1/f = 1/v - 1/u
where f is the focal length of the lens, v is the image distance, and u is the object distance.

Step 3: Calculate the image distance
Given: u = -10 cm (negative sign indicates that the object is placed on the left side of the lens combination)

Using the magnification formula:
m = v/u
3 = v / -10
v = -30 cm (negative sign indicates that the image is formed on the left side of the lens combination)

Step 4: Find the focal length of lens A
Since the image is formed on the same side as lens B, the combination of lenses acts as a microscope.

Using the lens formula for the combination of lenses:
1/f = 1/f1 + 1/f2
where f1 and f2 are the focal lengths of lenses A and B, respectively.

Given: f2 = -30 cm (negative sign indicates a concave lens)

Using the lens formula:
1/f = 1/f1 - 1/30
Since the combination acts as a microscope, the focal length of lens A should be positive.

Simplifying the equation:
1/f = (30 - f1) / (30*f1)
30*f1 = 30 - f1
31*f1 = 30
f1 = 30/31 ≈ 0.97 cm

Step 5: Determine the nature of lens A
Since the focal length of lens A is positive, it is a convex lens.

Therefore, the correct answer is option 'D' - Convex, 6 cm.

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