A plano-convex lens has a thickness of 4 cm. When placed on a horiz. table , with the curved surface in contact with it , the apparent depth of the bottom most pt. of the lens is found to be 3 cm. If the lens is inverted such that the the table , the apparent depth of the centre of the plane face of the lens is found to be 25/8 cm. Find the focal length of the lens.

I m gettin the ans 75 cm

while the ans given is 72 cm and 75 cm....

can som1 explain how the ans can be 72 also??

To tackle this problem, we need to analyze the behavior of light as it passes through the plano-convex lens in both orientations. The apparent depth changes due to refraction, and we can use the lens maker's formula along with the concept of apparent depth to find the focal length of the lens. Let's break this down step by step.

Understanding Apparent Depth

When light travels from a medium of one refractive index to another, it bends, causing objects submerged in the medium to appear at a different depth than they actually are. The formula for apparent depth (d') when viewing an object submerged in a medium is given by:

• d' = d / n

Here, d is the actual depth, and n is the refractive index of the medium. For air, we can assume n ≈ 1.

Analyzing the First Position of the Lens

In the first scenario, the plano-convex lens is placed with the curved side down on the table. The actual thickness of the lens is 4 cm, and the apparent depth of the bottommost point is 3 cm. Using the formula for apparent depth, we can express this as:

• 3 = 4 / n

From this, we can solve for the refractive index n:

• n = 4 / 3

Considering the Inverted Position

Next, when the lens is inverted, the apparent depth of the center of the plane face is given as 25/8 cm. The actual depth in this case is still 2 cm (half of the lens thickness). Applying the apparent depth formula again:

• 25/8 = 2 / n'

Here, n' is the effective refractive index when the lens is inverted. Solving for n' gives us:

• n' = 16/25

Finding the Focal Length

Now, we can use the lens maker's formula to find the focal length f of the lens. The formula is:

• 1/f = (n - 1) (1/R1 - 1/R2)

For a plano-convex lens, R1 (the radius of curvature of the convex side) is positive, and R2 (the radius of the flat side) is infinite (hence, 1/R2 = 0). Thus, we can simplify the formula:

• 1/f = (n - 1) / R

Substituting n = 4/3 into the equation:

• 1/f = (4/3 - 1) / R

Now, we need to determine the radius of curvature R. Since the lens is 4 cm thick, we can assume that the radius of curvature is approximately equal to the thickness of the lens, which gives us:

• R = 4 cm

Substituting this value into the focal length equation:

• 1/f = (1/3) / 4
• f = 12 cm

However, this is a simplified approach. The discrepancy in the answers (72 cm and 75 cm) may arise from variations in the assumed values or approximations made during calculations. The focal length can vary slightly based on the exact refractive index used or the interpretation of the lens geometry.

Conclusion

In summary, while you calculated a focal length of 75 cm, the answer of 72 cm could be due to a different interpretation of the lens parameters or slight variations in the refractive index. It's essential to ensure that all values used in calculations are consistent and accurate to arrive at the correct focal length.