If in a planoconvex lens, the radius of curvature of the convex surface is `10cm` and the focal length is `30 cm`, the refractive index of the material of the lens will be

Options

A. 1.5
B. 1.66
C. 1.33
D. 3

Answer

To find the refractive index (μ) of a planoconvex lens given the radius of curvature (R) and the focal length (f), we can use the lens maker's formula:

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

For a planoconvex lens:
- The radius of curvature of the convex surface (R1) is positive: \( R_1 = +10 \, \text{cm} \)
- The radius of curvature of the flat surface (R2) is considered infinite: \( R_2 = -\infty \)
- The focal length (f) is given as \( +30 \, \text{cm} \)

### Step 1: Substitute the values into the lens maker's formula

Using the values provided:
\[ \frac{1}{30} = \left( \mu - 1 \right) \left( \frac{1}{10} - 0 \right) \]

Since \( \frac{1}{R_2} = 0 \) (because \( R_2 = -\infty \)), the equation simplifies to:
\[ \frac{1}{30} = \left( \mu - 1 \right) \frac{1}{10} \]

### Step 2: Rearrange the equation to solve for (μ - 1)

Multiply both sides by 10:
\[ \frac{10}{30} = \mu - 1 \]
\[ \frac{1}{3} = \mu - 1 \]

### Step 3: Solve for μ

Add 1 to both sides:
\[ \mu = 1 + \frac{1}{3} \]
\[ \mu = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

### Conclusion

Thus, the refractive index (μ) of the material of the lens is:
\[ \mu = \frac{4}{3} \]