A double convex lens has power P and same radii of curvature R of both the surfaces. The radius of carvature of a surface of plano-convex lens made of the same meterial with power 1.5 P is :

Options

A. 2R
B. `R/2`
C. `(3R)/2`
D. `R/3`

Answer

To solve the problem, we need to find the radius of curvature of a surface of a plano-convex lens made of the same material as a double convex lens with given powers.

### Step-by-Step Solution:

1. **Understanding the Power of the Double Convex Lens**:
- The power \( P \) of a lens is given by the formula:
\[ P = \frac{1}{F} \]
- The lensmaker's formula for a double convex lens with equal radii of curvature \( R \) is:
\[ \frac{1}{F} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- For a double convex lens, we have \( R_1 = R \) and \( R_2 = -R \):
\[ \frac{1}{F} = (n - 1) \left( \frac{1}{R} - \frac{-1}{R} \right) = (n - 1) \left( \frac{2}{R} \right) \]
- Therefore, we can express the power \( P \) as:
\[ P = (n - 1) \frac{2}{R} \]

2. **Finding the Power of the Plano-Convex Lens**:
- The power of the plano-convex lens is given as \( 1.5P \).
- For the plano-convex lens, we have \( R_1 = R_1' \) (the radius we need to find) and \( R_2 = \infty \):
\[ \frac{1}{F'} = (n - 1) \left( \frac{1}{R_1'} - 0 \right) = (n - 1) \frac{1}{R_1'} \]
- The power \( P' \) of the plano-convex lens is:
\[ P' = (n - 1) \frac{1}{R_1'} \]
- Setting this equal to \( 1.5P \):
\[ (n - 1) \frac{1}{R_1'} = 1.5(n - 1) \frac{2}{R} \]

3. **Solving for \( R_1' \)**:
- We can simplify the equation:
\[ \frac{1}{R_1'} = 1.5 \cdot \frac{2}{R} \]
- This leads to:
\[ R_1' = \frac{R}{3} \]

### Final Answer:
The radius of curvature of the surface of the plano-convex lens is:
\[ R_1' = \frac{R}{3} \]