Short Answer 
The lens maker’s formula, ( frac{1}{f} = (n – 1) left( frac{1}{R_1} – frac{1}{R_2} right) ), relates a lens’s focal length and radii of curvature to its refractive index. By substituting specific values and simplifying, we find that the refractive index (n) equals (frac{3}{2}).
Step 1: Understand the Lens Maker’s Formula
The lens maker’s formula is essential for determining the refractive index of a lens material. It establishes a relationship between the focal length (f) of the lens and its radii of curvature (R1 and R2). The formula is given by:
- (frac{1}{f} = (n – 1) left( frac{1}{R_1} – frac{1}{R_2} right))
Here, (n) represents the refractive index of the lens material. Understanding this formula is crucial for the calculation process.
Step 2: Substitute Given Values into the Formula
To find the refractive index, we need specific values. For a lens with radii of curvature of R and 2R, and a focal length of (frac{4}{3}R), we substitute these into the lens maker’s formula:
- (frac{1}{frac{4}{3}R} = (n – 1) left( frac{1}{R} – frac{1}{2R} right))
Simplifying this equation is a critical next step to isolate (n) for the calculation of the refractive index.
Step 3: Solve for the Refractive Index
After simplification, we arrive at the equation:
- (frac{3}{4R} = (n – 1) left( frac{1}{2R} right))
Multiplying both sides by (2R) gives:
- (frac{3}{2} = n – 1)
By adding 1 to both sides, we find that the refractive index (n) equals (frac{3}{2}), confirming the calculation.