The radii of curvatures of a double convex lens are 15 cm and 30 cm, and its refractive index is 1.5. Then its focal length is -

Options

A. `+ 10 cm `
B. `- 10 cm `
C. `+ 20 cm `
D. `- 20 cm `

Answer

To find the focal length of a double convex lens given the radii of curvature and refractive index, we can use the lensmaker's formula:

\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Where:
- \( f \) is the focal length,
- \( n \) is the refractive index,
- \( R_1 \) is the radius of curvature of the first surface,
- \( R_2 \) is the radius of curvature of the second surface.

### Step 1: Identify the values
Given:
- \( R_1 = 15 \, \text{cm} \)
- \( R_2 = -30 \, \text{cm} \) (the second radius is negative because it is a convex lens)
- \( n = 1.5 \)

### Step 2: Substitute the values into the lensmaker's formula
Using the lensmaker's formula:

\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{15} - \frac{1}{-30} \right) \]

### Step 3: Simplify the expression
Calculate \( n - 1 \):

\[ n - 1 = 1.5 - 1 = 0.5 \]

Now calculate \( \frac{1}{15} - \frac{1}{-30} \):

\[ \frac{1}{15} + \frac{1}{30} = \frac{2}{30} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10} \]

### Step 4: Substitute back into the formula
Now substitute back into the formula:

\[ \frac{1}{f} = 0.5 \times \frac{1}{10} \]

\[ \frac{1}{f} = \frac{0.5}{10} = \frac{1}{20} \]

### Step 5: Calculate the focal length
Now, taking the reciprocal to find \( f \):

\[ f = 20 \, \text{cm} \]

Thus, the focal length of the lens is \( 20 \, \text{cm} \).

### Final Answer:
The focal length of the double convex lens is \( 20 \, \text{cm} \).

---