The radii of curvature of the two surfaces of a lens are `20cm` and `30 cm` and the refractive index of the material of the lens is `1.5`. If the lens is concave`-` convex, then the focal length of lens is

Options

A. `24 cm`
B. `10 cm`
C. `15 cm`
D. `120 cm`

Explanation

To find the focal length of a concave-convex lens using the given data, we can apply the lens maker's formula. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Radius of curvature of the first surface, \( R_1 = -20 \, \text{cm} \) (concave surface is taken as negative)
- Radius of curvature of the second surface, \( R_2 = +30 \, \text{cm} \) (convex surface is taken as positive)
- Refractive index of the lens material, \( \mu = 1.5 \)

### Step 2: Write down the lens maker's formula
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: Substitute the values into the formula
Substituting the known values into the formula:
\[ \frac{1}{F} = (1.5 - 1) \left( \frac{1}{-20} - \frac{1}{30} \right) \]
This simplifies to:
\[ \frac{1}{F} = 0.5 \left( \frac{1}{-20} - \frac{1}{30} \right) \]

### Step 4: Calculate the terms inside the parentheses
To combine the fractions:
\[ \frac{1}{-20} - \frac{1}{30} = \frac{-3 + 2}{60} = \frac{-1}{60} \]

### Step 5: Substitute back into the formula
Now substituting back, we have:
\[ \frac{1}{F} = 0.5 \left( \frac{-1}{60} \right) = \frac{-0.5}{60} = \frac{-1}{120} \]

### Step 6: Solve for F
Taking the reciprocal gives:
\[ F = -120 \, \text{cm} \]

### Step 7: Interpret the result
The negative sign indicates that the focal length is on the same side as the incoming light, which is characteristic of a concave lens. Therefore, the magnitude of the focal length is:
\[ |F| = 120 \, \text{cm} \]

### Final Answer
The focal length of the lens is \( 120 \, \text{cm} \).

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