Diameter of the flat surface of a circular plano-convex lens is `6 cm` and thickness at the centre is `3 mm.` The radius of curvature of the curved part is

Options

A. `15 cm`
B. `20 cm`
C. `30 cm`
D. `10 cm`

Explanation

To find the radius of curvature (R) of the curved part of a circular plano-convex lens, we can follow these steps:

### Step 1: Understand the dimensions given
- The diameter of the flat surface of the lens is given as 6 cm.
- The thickness at the center (t) is given as 3 mm, which we need to convert to centimeters for consistency.

### Step 2: Convert the thickness to centimeters
- Since 1 cm = 10 mm, we convert 3 mm to centimeters:
\[ t = \frac{3 \text{ mm}}{10} = 0.3 \text{ cm} \]

### Step 3: Calculate the radius of the flat surface
- The radius (d) of the flat surface is half of the diameter:
\[ d = \frac{6 \text{ cm}}{2} = 3 \text{ cm} \]

### Step 4: Apply the relationship using the geometry of the lens
- For a plano-convex lens, we can use the Pythagorean theorem in the right triangle formed by the radius of curvature (R), the thickness (t), and the radius (d):
\[ R^2 = (R - t)^2 + d^2 \]

### Step 5: Expand and rearrange the equation
- Expanding the equation:
\[ R^2 = (R^2 - 2Rt + t^2) + d^2 \]
- Rearranging gives:
\[ 0 = -2Rt + t^2 + d^2 \]
- Rearranging further gives:
\[ 2Rt = t^2 + d^2 \]

### Step 6: Solve for R
- Dividing both sides by 2t:
\[ R = \frac{t^2 + d^2}{2t} \]

### Step 7: Substitute the values
- Substitute \(t = 0.3 \text{ cm}\) and \(d = 3 \text{ cm}\):
\[ R = \frac{(0.3)^2 + (3)^2}{2 \times 0.3} \]
\[ R = \frac{0.09 + 9}{0.6} \]
\[ R = \frac{9.09}{0.6} = 15.15 \text{ cm} \]

### Step 8: Round the answer
- The radius of curvature \(R\) can be approximated to \(15 \text{ cm}\).

### Final Answer
The radius of curvature of the curved part of the plano-convex lens is approximately **15 cm**.

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