To find the image distance for a biconvex lens formed by two planoconvex lenses, we can use the lens maker's formula and the lens formula. Let's break this down step by step.
Understanding the Lens System
A biconvex lens is essentially two planoconvex lenses joined together. In this case, we have:
- First lens with a refractive index (n1) of 1.5
- Second lens with a refractive index (n2) of 1.2
- Both lenses have the same radius of curvature (R) of 14 cm.
Using the Lens Formula
The lens formula is given by:
1/f = 1/v - 1/u
Where:
- f = focal length of the lens
- v = image distance
- u = object distance (which is -40 cm, as per the sign convention)
Calculating the Focal Length
For a lens, the focal length (f) can be calculated using the lens maker's formula:
f = (n - 1) * (R / 2)
We need to calculate the effective focal length of the biconvex lens, which is the combination of the two lenses.
Focal Length of Each Lens
For the first lens (n1 = 1.5):
f1 = (1.5 - 1) * (14 / 2) = 0.5 * 7 = 3.5 cm
For the second lens (n2 = 1.2):
f2 = (1.2 - 1) * (14 / 2) = 0.2 * 7 = 1.4 cm
Combining the Focal Lengths
Since the lenses are in contact, the effective focal length (f) of the biconvex lens can be found using the formula for two thin lenses in contact:
1/f = 1/f1 + 1/f2
Substituting the values:
1/f = 1/3.5 + 1/1.4
Calculating the right side:
1/f = 0.2857 + 0.7143 = 1
Thus, f = 1 cm.
Finding the Image Distance
Now, we can use the lens formula to find the image distance (v):
1/f = 1/v - 1/u
Substituting the known values:
1/1 = 1/v - 1/(-40)
Rearranging gives:
1/v = 1 + 1/40
1/v = 1 + 0.025 = 1.025
Thus, v = 1/1.025 ≈ 0.975 cm.
Final Result
The image distance for the biconvex lens, when the object is placed at a distance of 40 cm, is approximately 0.975 cm from the lens. This indicates that the image is formed on the same side as the object, which is typical for converging lenses when the object is placed beyond the focal length.
