The radii of curvature of the two surface of a double convex glass lens are 10 cm and 20 cm respectively. If the refractive index of glass in 1.5, then its focal length will be

Options

A. 8.5 cm
B. 13.33 cm
C. 20.7 cm
D. 25.2 cm

Explanation

To find the focal length of a double convex lens using the lens maker's formula, we can follow these steps:

### Step 1: Understand the Lens Maker's Formula
The lens maker's formula is given by:
\[ \frac{1}{f} = \left( n - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where:
- \( f \) is the focal length of the lens,
- \( n \) is the refractive index of the lens material,
- \( R_1 \) is the radius of curvature of the first surface,
- \( R_2 \) is the radius of curvature of the second surface.

### Step 2: Assign Values
From the problem, we have:
- \( R_1 = 10 \, \text{cm} \) (positive for a convex surface),
- \( R_2 = 20 \, \text{cm} \) (negative for a convex surface, according to the sign convention),
- \( n = 1.5 \).

### Step 3: Substitute Values into the Formula
Substituting the values into the lens maker's formula:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - \frac{1}{-20} \right) \]

### Step 4: Simplify the Equation
Calculating the terms:
\[ \frac{1}{f} = 0.5 \left( \frac{1}{10} + \frac{1}{20} \right) \]
Finding a common denominator for the fractions inside the parentheses:
\[ \frac{1}{10} = \frac{2}{20} \]
So,
\[ \frac{1}{f} = 0.5 \left( \frac{2}{20} + \frac{1}{20} \right) = 0.5 \left( \frac{3}{20} \right) \]

### Step 5: Final Calculation
Now, simplifying further:
\[ \frac{1}{f} = \frac{1.5}{20} \]
Thus,
\[ f = \frac{20}{3} \approx 6.67 \, \text{cm} \]

### Final Answer
The focal length of the lens is approximately \( 6.67 \, \text{cm} \).

---