The refractive index of the material ofa double convex lens is 1.5 and its focal length is 5 cm. If the radii of curvature are equal, the value of the radius of curvature (in cm) is

Options

A. 5
B. 6.5
C. 8
D. 9.5

Answer

To solve the problem, we need to find the radius of curvature (r) of a double convex lens given its refractive index (μ) and focal length (f). The steps to derive the solution are as follows:

### Step 1: Identify the given values
- Refractive index (μ) = 1.5
- Focal length (f) = 5 cm

### Step 2: Write the lens maker's formula
For a double convex lens, the lens maker's formula is given by:

\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

### Step 3: Assign values to the radii of curvature
Since the radii of curvature are equal, we can denote them as:
- \( r_1 = r \)
- \( r_2 = -r \) (the negative sign indicates that the second radius is in the opposite direction)

### Step 4: Substitute the values into the formula
Substituting \( r_1 \) and \( r_2 \) into the lens maker's formula gives:

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

This simplifies to:

\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{r} + \frac{1}{r} \right) = (\mu - 1) \left( \frac{2}{r} \right) \]

### Step 5: Substitute the known values
Now substituting \( \mu = 1.5 \) and \( f = 5 \) cm into the equation:

\[ \frac{1}{5} = (1.5 - 1) \left( \frac{2}{r} \right) \]

This simplifies to:

\[ \frac{1}{5} = 0.5 \left( \frac{2}{r} \right) \]

### Step 6: Solve for r
Now, we can solve for \( r \):

\[ \frac{1}{5} = \frac{1}{r} \]

Cross-multiplying gives:

\[ r = 5 \text{ cm} \]

### Final Answer
The radius of curvature \( r \) is 5 cm.

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