The focal length of a convex lens is 10 cm and its refractive index is 1.5. If the radius of curvature of one surface is 7.5 cm , the radius of curvature of the second surface will be

Options

A. 7.5 cm
B. 15.0 cm
C. 75 cm
D. 5.0 cm

Answer

To find the radius of curvature of the second surface of a convex lens, we can use the Lensmaker's formula, which is given by:

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

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

### Step 1: Identify the known values
- Focal length \( F = 10 \, \text{cm} \) (positive for a convex lens)
- Refractive index \( \mu = 1.5 \)
- Radius of curvature of the first surface \( R_1 = 7.5 \, \text{cm} \) (positive for convex)

### Step 2: Write the Lensmaker's formula
Substituting the known values into the Lensmaker's formula:

\[ \frac{1}{10} = (1.5 - 1) \left( \frac{1}{7.5} - \frac{1}{R_2} \right) \]

### Step 3: Simplify the equation
Calculating \( \mu - 1 \):

\[ 1.5 - 1 = 0.5 \]

So the equation becomes:

\[ \frac{1}{10} = 0.5 \left( \frac{1}{7.5} - \frac{1}{R_2} \right) \]

### Step 4: Solve for \( \frac{1}{R_2} \)
First, multiply both sides by 2 to eliminate the 0.5:

\[ \frac{2}{10} = \frac{1}{7.5} - \frac{1}{R_2} \]

This simplifies to:

\[ \frac{1}{5} = \frac{1}{7.5} - \frac{1}{R_2} \]

### Step 5: Find a common denominator
The common denominator for \( \frac{1}{5} \) and \( \frac{1}{7.5} \) is \( 37.5 \):

\[ \frac{1}{5} = \frac{7.5}{37.5} \quad \text{and} \quad \frac{1}{7.5} = \frac{5}{37.5} \]

So we rewrite the equation:

\[ \frac{7.5}{37.5} = \frac{5}{37.5} - \frac{1}{R_2} \]

### Step 6: Rearranging the equation
Rearranging gives:

\[ \frac{1}{R_2} = \frac{5}{37.5} - \frac{7.5}{37.5} \]

This simplifies to:

\[ \frac{1}{R_2} = \frac{-2.5}{37.5} \]

### Step 7: Calculate \( R_2 \)
Taking the reciprocal gives:

\[ R_2 = -15 \, \text{cm} \]

### Conclusion
The radius of curvature of the second surface is \( R_2 = -15 \, \text{cm} \).