In Y.D.S.E. , distance between both the slits is 2 mm. When a light of frequency `5.5 xx 10^(14) Hz` is used, the fringes width is 1 mm. Determine the distance between screen and slit.

Answer

To determine the distance between the screen and the slits in a Young's Double Slit Experiment (Y.D.S.E), we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Distance between slits (d) = 2 mm = \(2 \times 10^{-3}\) m
- Frequency of light (f) = \(5.5 \times 10^{14}\) Hz
- Fringe width (β) = 1 mm = \(1 \times 10^{-3}\) m

2. **Calculate the Wavelength (λ)**:
- Use the formula relating speed of light (c), frequency (f), and wavelength (λ):
\[ c = f \lambda \]
- The speed of light (c) is approximately \(3 \times 10^8\) m/s.
- Rearranging the formula to find λ:
\[ \lambda = \frac{c}{f} \]
- Substitute the values:
\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{5.5 \times 10^{14} \text{ Hz}} \approx 5.45 \times 10^{-7} \text{ m} \]

3. **Use the Fringe Width Formula**:
- The fringe width (β) is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
- Where D is the distance between the screen and the slits, and d is the distance between the slits.

4. **Rearranging the Formula to Find D**:
- Rearranging the formula to solve for D:
\[ D = \frac{\beta d}{\lambda} \]

5. **Substituting the Values**:
- Substitute the known values into the equation:
\[ D = \frac{(1 \times 10^{-3} \text{ m})(2 \times 10^{-3} \text{ m})}{5.45 \times 10^{-7} \text{ m}} \]
- Calculate D:
\[ D = \frac{2 \times 10^{-6}}{5.45 \times 10^{-7}} \approx 3.67 \text{ m} \]

6. **Final Answer**:
- The distance between the screen and the slits (D) is approximately **3.67 m**.