A slit of aperture 1.8 mm is illuminated by light of wavelength `5,000 Å`. Calculate the minimum distance light can travel before the diffraction is noticed.

Answer

To solve the problem of calculating the minimum distance light can travel before diffraction is noticed, we will use the concept of Fresnel distance. The Fresnel distance (ZF) is given by the formula:

\[ Z_F = \frac{D^2}{\lambda} \]

Where:
- \( D \) is the aperture of the slit (in meters)
- \( \lambda \) is the wavelength of the light (in meters)

### Step 1: Convert the given values to appropriate units
- The aperture \( D = 1.8 \, \text{mm} = 1.8 \times 10^{-3} \, \text{m} \)
- The wavelength \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5.0 \times 10^{-7} \, \text{m} \)

### Step 2: Substitute the values into the Fresnel distance formula
Now we can substitute the values of \( D \) and \( \lambda \) into the Fresnel distance formula:

\[ Z_F = \frac{(1.8 \times 10^{-3})^2}{5.0 \times 10^{-7}} \]

### Step 3: Calculate \( D^2 \)
Calculate \( D^2 \):

\[ D^2 = (1.8 \times 10^{-3})^2 = 3.24 \times 10^{-6} \, \text{m}^2 \]

### Step 4: Substitute \( D^2 \) into the formula
Now substitute \( D^2 \) into the formula:

\[ Z_F = \frac{3.24 \times 10^{-6}}{5.0 \times 10^{-7}} \]

### Step 5: Perform the division
Now, perform the division:

\[ Z_F = \frac{3.24}{5.0} \times 10^{1} = 0.648 \times 10^{1} = 6.48 \, \text{m} \]

### Conclusion
The minimum distance light can travel before diffraction is noticed is:

\[ Z_F = 6.48 \, \text{m} \]

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