The fringe width in YDSE is `2.4 xx 10^(-4)m`, when red light of wavelength `6400 Å` is used. By how much will it change, if blue light of wavelength `4000 Å` is used ?

Explanation

To solve the problem, we will use the formula for fringe width in Young's Double Slit Experiment (YDSE):

\[ \text{Fringe Width} (x) = \frac{\lambda D}{d} \]

Where:
- \( \lambda \) = wavelength of light
- \( D \) = distance between the slits and the screen
- \( d \) = distance between the slits

Given:
- Fringe width with red light (\( x_1 \)) = \( 2.4 \times 10^{-4} \) m
- Wavelength of red light (\( \lambda_1 \)) = \( 6400 \) Å = \( 6400 \times 10^{-10} \) m
- Wavelength of blue light (\( \lambda_2 \)) = \( 4000 \) Å = \( 4000 \times 10^{-10} \) m

### Step 1: Write the relationship between fringe widths and wavelengths

From the formula for fringe width, we can express the fringe widths for both wavelengths:

\[ x_1 = \frac{\lambda_1 D}{d} \]
\[ x_2 = \frac{\lambda_2 D}{d} \]

### Step 2: Set up the ratio of fringe widths

Taking the ratio of the fringe widths:

\[ \frac{x_1}{x_2} = \frac{\lambda_1}{\lambda_2} \]

### Step 3: Substitute the known values

Substituting the known values into the ratio:

\[ \frac{2.4 \times 10^{-4}}{x_2} = \frac{6400 \times 10^{-10}}{4000 \times 10^{-10}} \]

### Step 4: Simplify the right side

Calculating the right side:

\[ \frac{6400}{4000} = 1.6 \]

So we have:

\[ \frac{2.4 \times 10^{-4}}{x_2} = 1.6 \]

### Step 5: Solve for \( x_2 \)

Now, we can solve for \( x_2 \):

\[ x_2 = \frac{2.4 \times 10^{-4}}{1.6} \]

Calculating \( x_2 \):

\[ x_2 = 1.5 \times 10^{-4} \, \text{m} \]

### Step 6: Calculate the change in fringe width

Now, we need to find the change in fringe width:

Change in fringe width = \( x_1 - x_2 \)

Substituting the values:

\[ \text{Change} = 2.4 \times 10^{-4} - 1.5 \times 10^{-4} \]

Calculating the change:

\[ \text{Change} = 0.9 \times 10^{-4} \, \text{m} \]

### Final Answer

The change in fringe width when switching from red light to blue light is:

\[ 0.9 \times 10^{-4} \, \text{m} \]

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