Answer:
The position of the second lens is approximately -41.16 cm from the first lens (behind it), and the focal length of the second lens is approximately 2.67 cm.
Step-by-step explanation:
To solve this problem, we can use the lens maker's formula and the lens formula. The lens maker's formula relates the focal length of a lens to the refractive indices of the lens material and the radii of curvature of its surfaces. The lens formula gives the relationship between the object distance (u), the image distance (v), and the focal length (f) of a lens.
Let's denote the object distance from the first lens as u1, the image distance from the first lens as v1, and the focal length of the second lens as f2.
Given data:
Object distance (u1) = 8 cm
Focal length of the first lens (f1) = 6 cm
Magnification (M) = (2.4 cm) / (0.2 cm) = 12
(Magnification M = -(v1/u1) = -height of the image / height of the object)
Now, let's start the calculations:
Step 1: Calculate the image distance (v1) formed by the first lens using the lens formula:
(1/f1) = (1/v1) - (1/u1)
(1/6) = (1/v1) - (1/8)
(1/v1) = (1/6) + (1/8)
(1/v1) = (4 + 3) / 24
(1/v1) = 7 / 24
v1 = 24 / 7 cm ≈ 3.43 cm
Step 2: Calculate the object distance (u2) for the second lens:
Since the final image coincides with the scale itself, the object distance for the second lens is the image distance (v1) of the first lens.
u2 = v1 = 3.43 cm
Step 3: Calculate the image distance (v2) formed by the second lens using the lens formula:
(1/f2) = (1/v2) - (1/u2)
(1/f2) = (1/v2) - (1/3.43)
(1/v2) = (1/f2) + (1/3.43)
Step 4: Use the given magnification (M) and the lens formula to find the value of v2:
M = -(v2/u2) [Negative sign indicates an inverted image]
12 = -(v2/3.43)
v2 = -(12 * 3.43) ≈ -41.16 cm
Step 5: Find the focal length (f2) of the second lens using the lens formula:
(1/f2) = (1/v2) - (1/u2)
(1/f2) = (1/-41.16) - (1/3.43)
(1/f2) = (-1/41.16) - (1/3.43)
(1/f2) = (-3.43 - 12) / 41.16
(1/f2) = -15.43 / 41.16
f2 ≈ -41.16 / -15.43
f2 ≈ 2.67 cm
Therefore, the position of the second lens is approximately -41.16 cm from the first lens (behind it), and the focal length of the second lens is approximately 2.67 cm. Note that the negative sign for the image distance (v2) and the focal length (f2) indicates that the image formed by the second lens is virtual and on the same side as the object (i.e., it is an erect image).