The radius curvature of each surface of a convex lens of refractive index 1.5 is 40 cm. Calculate its power.

Options

A. `2.5 D`
B. `2 D`
C. `1.5 D`
D. `1 D`

Explanation

To solve the problem of calculating the power of a convex lens with a given refractive index and radius of curvature, we will use the lens maker's formula. Here are the steps:

### Step 1: Understand the Given Data
- Refractive index (N) of the lens = 1.5
- Radius of curvature of each surface (R1 and R2) = 40 cm

### Step 2: Convert Units
Since the lens maker's formula requires the radius of curvature to be in meters, we need to convert 40 cm to meters:
- R1 = +40 cm = +0.4 m
- R2 = -40 cm = -0.4 m (the negative sign indicates that the second surface is concave with respect to the incoming light)

### Step 3: Write the Lens Maker's Formula
The lens maker's formula is given by:
\[ \frac{1}{F} = (N - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where \( F \) is the focal length of the lens.

### Step 4: Substitute the Values into the Formula
Substituting the values into the lens maker's formula:
\[ \frac{1}{F} = (1.5 - 1) \left( \frac{1}{0.4} - \frac{1}{-0.4} \right) \]
Calculating \( N - 1 \):
\[ N - 1 = 1.5 - 1 = 0.5 \]
Now calculate \( \frac{1}{0.4} - \frac{1}{-0.4} \):
\[ \frac{1}{0.4} - \frac{1}{-0.4} = \frac{1}{0.4} + \frac{1}{0.4} = 2 \cdot \frac{1}{0.4} = \frac{2}{0.4} = 5 \]

### Step 5: Calculate \( \frac{1}{F} \)
Now substituting back into the equation:
\[ \frac{1}{F} = 0.5 \cdot 5 = 2.5 \]

### Step 6: Calculate the Power of the Lens
The power \( P \) of the lens is given by:
\[ P = \frac{1}{F} \]
Thus, the power of the lens is:
\[ P = 2.5 \text{ diopters} \]

### Final Answer
The power of the convex lens is **2.5 diopters**.

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