The radii of curvatures of a convex lens are 0.04 m and 0.04m. Its refractive index is 1.5. Its focal length is

Options

A. 0.04 m
B. 0.4 m
C. 4m
D. 40 m

Answer

To find the focal length of the convex lens, we can use the lens maker's formula, which is given by:

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

Where:
- \( f \) is the focal length of the lens,
- \( \mu \) 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 1: Identify the given values
- The radii of curvature \( R_1 = 0.04 \, \text{m} \) (for the first surface) and \( R_2 = 0.04 \, \text{m} \) (for the second surface).
- The refractive index \( \mu = 1.5 \).

### Step 2: Apply the lens maker's formula
Using the lens maker's formula, we can substitute the values we have:

\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{0.04} - \frac{1}{-0.04} \right) \]

### Step 3: Calculate the relative refractive index
Calculate \( \mu - 1 \):

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

### Step 4: Calculate the terms involving radii of curvature
Now calculate the terms involving \( R_1 \) and \( R_2 \):

\[ \frac{1}{R_1} - \frac{1}{R_2} = \frac{1}{0.04} - \frac{1}{-0.04} \]

### Step 5: Simplify the expression
Calculating the above expression:

\[ \frac{1}{0.04} - \left(-\frac{1}{0.04}\right) = \frac{1}{0.04} + \frac{1}{0.04} = 2 \cdot \frac{1}{0.04} = \frac{2}{0.04} \]

### Step 6: Substitute back into the formula
Now substitute this back into the lens maker's formula:

\[ \frac{1}{f} = 0.5 \cdot \frac{2}{0.04} \]

### Step 7: Calculate \( \frac{1}{f} \)
Calculating \( 0.5 \cdot \frac{2}{0.04} \):

\[ \frac{1}{f} = \frac{1}{0.04} = 25 \]

### Step 8: Find \( f \)
Taking the reciprocal to find \( f \):

\[ f = 0.04 \, \text{m} \]

### Conclusion
Thus, the focal length of the convex lens is:

\[ \boxed{0.04 \, \text{m}} \]