The refractive index of a material of a plano-concave lens is `5/3`, the radius of curvature is 0.3 m. The focal length of the lens in air is

Options

A. `-0.45m`
B. `-0.6m`
C. `-0.75m`
D. `-1.0m`

Answer

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

### Step 1: Understand the given data
- Refractive index (μ) = \( \frac{5}{3} \)
- Radius of curvature (R) = 0.3 m
- For a plano-concave lens, one surface is flat (plane) and the other is concave.

### Step 2: Apply the lens maker's formula
The lens maker's formula for a lens is given by:

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

Where:
- \( f \) = focal length
- \( \mu \) = refractive index of the lens material
- \( R_1 \) = radius of curvature of the first surface
- \( R_2 \) = radius of curvature of the second surface

### Step 3: Assign values to \( R_1 \) and \( R_2 \)
For a plano-concave lens:
- The flat surface (plane) has an infinite radius of curvature, so \( R_1 = \infty \).
- The concave surface has a radius of curvature \( R_2 = -0.3 \) m (negative because it is concave).

### Step 4: Substitute values into the formula
Substituting the values into the lens maker's formula:

\[ \frac{1}{f} = \left( \frac{5}{3} - 1 \right) \left( \frac{1}{\infty} - \frac{1}{-0.3} \right) \]

### Step 5: Simplify the equation
Calculating \( \mu - 1 \):

\[ \mu - 1 = \frac{5}{3} - 1 = \frac{5}{3} - \frac{3}{3} = \frac{2}{3} \]

Now, substituting this back into the formula:

\[ \frac{1}{f} = \left( \frac{2}{3} \right) \left( 0 - \left( -\frac{1}{0.3} \right) \right) \]

This simplifies to:

\[ \frac{1}{f} = \left( \frac{2}{3} \right) \left( \frac{1}{0.3} \right) \]

### Step 6: Calculate \( \frac{1}{f} \)
Calculating \( \frac{1}{0.3} \):

\[ \frac{1}{0.3} = \frac{10}{3} \]

So,

\[ \frac{1}{f} = \frac{2}{3} \times \frac{10}{3} = \frac{20}{9} \]

### Step 7: Find \( f \)
Taking the reciprocal gives us the focal length:

\[ f = \frac{9}{20} \text{ m} \]

### Step 8: Final answer
Thus, the focal length of the plano-concave lens in air is:

\[ f = -0.45 \text{ m} \]

(The negative sign indicates that it is a diverging lens.)