To determine the radius of curvature of a diverging lens when given the object distance, image distance, and refractive index, we can use the lens maker's formula along with the lens formula. Let's break this down step by step.
Understanding the Problem
We have a diverging lens with the following parameters:
- Object distance (u) = -60 cm (negative because the object is on the same side as the incoming light)
- Image distance (v) = -20 cm (also negative for virtual images)
- Refractive index (n) = 1.65
Using the Lens Formula
The lens formula relates the object distance (u), image distance (v), and the focal length (f) of the lens:
1/f = 1/v - 1/u
Substituting the known values:
1/f = 1/(-20) - 1/(-60)
1/f = -1/20 + 1/60
To combine these fractions, we need a common denominator, which is 60:
1/f = -3/60 + 1/60 = -2/60 = -1/30
Thus, the focal length (f) is:
f = -30 cm
Applying the Lens Maker's Formula
The lens maker's formula is given by:
1/f = (n - 1) * (1/R1 - 1/R2)
For a diverging lens, R1 is positive (convex surface) and R2 is negative (concave surface). Let's denote R1 as R and R2 as -R for simplicity:
1/f = (n - 1) * (1/R - 1/(-R)) = (n - 1) * (1/R + 1/R) = (n - 1) * (2/R)
Substituting the values we have:
1/(-30) = (1.65 - 1) * (2/R)
-1/30 = 0.65 * (2/R)
Solving for Radius of Curvature
Now we can solve for R:
-1/30 = 1.3/R
Cross-multiplying gives:
-R = 30 * 1.3
-R = 39
Thus, R = -39 cm. The negative sign indicates that the radius of curvature is on the side of the incoming light, which is consistent with the nature of a diverging lens.
Final Result
The radius of curvature of the lens is approximately 39 cm.
