A biconvex lens of radii of curvature 20 cm has refractive index 1.56 for blue colour and 1.48 for red colour. The approximate linear spread of focus for white light is

Options

A. 3.6 cm
B. 4.2 cm
C. 3.0 cm
D. 5.0 cm

Explanation

To solve the problem of finding the approximate linear spread of focus for white light using a biconvex lens, we will follow these steps:

### Step 1: Understand the Lens Formula
The lens formula for a biconvex lens is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For a biconvex lens, both radii of curvature (R1 and R2) are equal in magnitude but opposite in sign. Thus, if we denote the radius of curvature as \( R \):
- \( R_1 = +R \)
- \( R_2 = -R \)

### Step 2: Substitute Values into the Lens Formula
Substituting the values into the lens formula:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R} - \frac{1}{-R} \right) = (\mu - 1) \left( \frac{1}{R} + \frac{1}{R} \right) = (\mu - 1) \left( \frac{2}{R} \right) \]
Thus, we can express the focal length \( f \) as:
\[ f = \frac{R}{2(\mu - 1)} \]

### Step 3: Calculate Focal Length for Blue and Red Light
Given:
- \( R = 20 \, \text{cm} \)
- \( \mu_{blue} = 1.56 \)
- \( \mu_{red} = 1.48 \)

Now, calculate the focal lengths for blue and red light:
1. For blue light:
\[ f_{blue} = \frac{20}{2(1.56 - 1)} = \frac{20}{2(0.56)} = \frac{20}{1.12} \approx 17.86 \, \text{cm} \]

2. For red light:
\[ f_{red} = \frac{20}{2(1.48 - 1)} = \frac{20}{2(0.48)} = \frac{20}{0.96} \approx 20.83 \, \text{cm} \]

### Step 4: Calculate the Difference in Focal Lengths
Next, we find the difference in focal lengths:
\[ \Delta f = f_{red} - f_{blue} \approx 20.83 - 17.86 \approx 2.97 \, \text{cm} \approx 3 \, \text{cm} \]

### Step 5: Conclusion
The approximate linear spread of focus for white light is:
\[ \Delta f \approx 3 \, \text{cm} \]