A single slit Fraunhofer diffraction pattern is formed with white light. For what wavelength of light the third secondary maximum in the diffraction pattern coincides with the secondary maximum in the pattern for red light of wavelength 6500 Å ?

Options

A. `9100Å`
B. `4642Å`
C. `4100Å`
D. `4400Å`

Answer

To solve the problem of finding the wavelength of light for which the third secondary maximum in the diffraction pattern coincides with the secondary maximum in the pattern for red light of wavelength 6500 Å, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Position of Secondary Maxima**:
The position of the nth secondary maximum in a single slit diffraction pattern is given by the formula:
\[ x_n = \frac{(2n + 1) \lambda D}{2a} \]
where:
- \( n \) = order of the secondary maximum
- \( \lambda \) = wavelength of light
- \( D \) = distance from the slit to the screen
- \( a \) = width of the slit

2. **Setting Up the Equation for Red Light**:
For red light with a wavelength of \( \lambda_{red} = 6500 \) Å, we want to find the position of the second secondary maximum (n=2):
\[ x_{red} = \frac{(2 \cdot 2 + 1) \cdot 6500 \cdot D}{2a} = \frac{5 \cdot 6500 \cdot D}{2a} \]

3. **Setting Up the Equation for the Unknown Wavelength**:
For the unknown wavelength \( \lambda_{unknown} \), we want to find the position of the third secondary maximum (n=3):
\[ x_{unknown} = \frac{(2 \cdot 3 + 1) \cdot \lambda_{unknown} \cdot D}{2a} = \frac{7 \cdot \lambda_{unknown} \cdot D}{2a} \]

4. **Equating the Two Positions**:
Since the third secondary maximum of the unknown light coincides with the second secondary maximum of the red light, we can set the two equations equal to each other:
\[ \frac{5 \cdot 6500 \cdot D}{2a} = \frac{7 \cdot \lambda_{unknown} \cdot D}{2a} \]

5. **Canceling Common Terms**:
We can cancel \( D \) and \( 2a \) from both sides:
\[ 5 \cdot 6500 = 7 \cdot \lambda_{unknown} \]

6. **Solving for the Unknown Wavelength**:
Rearranging the equation gives:
\[ \lambda_{unknown} = \frac{5 \cdot 6500}{7} \]
Now, calculating this:
\[ \lambda_{unknown} = \frac{32500}{7} \approx 4642.86 \text{ Å} \]

### Final Answer:
The wavelength of the unknown light is approximately **4642.86 Å**.