A double slit is illuminated with monochromatic light of wavelength `6.00 xx 10^(2)` nm. The m = 0 and m = 1 bright fringes are separated by 3.0cm on a screen which is located 4.0 m from the slits. What is the separation between the slits?

Options

A. `4.0 xx 10^(-5)` m
B. `1.2 xx 10^(-4)` m
C. `8.0 xx 10^(-5)` m
D. `1.6 xx 10^(-4)` m

Answer

To solve the problem, we need to find the separation between the slits (denoted as \(d\)) in a double-slit experiment. We are given the following information:

- Wavelength of light, \(\lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m}\)
- Distance between the m=0 and m=1 bright fringes, \(\Delta x = 3 \, \text{cm} = 3 \times 10^{-2} \, \text{m}\)
- Distance from the slits to the screen, \(D = 4 \, \text{m}\)

### Step-by-step Solution:

1. **Understand the relationship between fringe separation and slit separation**:
The distance between the m=0 and m=1 bright fringes on the screen can be expressed in terms of the wavelength and the slit separation using the formula:
\[ \Delta x = \frac{\lambda D}{d} \]
where \(\Delta x\) is the distance between the fringes, \(\lambda\) is the wavelength of the light, \(D\) is the distance from the slits to the screen, and \(d\) is the separation between the slits.

2. **Rearranging the formula to find \(d\)**:
We can rearrange the formula to solve for \(d\):
\[ d = \frac{\lambda D}{\Delta x} \]

3. **Substituting the known values**:
Now, we can substitute the known values into the equation:
- \(\lambda = 600 \times 10^{-9} \, \text{m}\)
- \(D = 4 \, \text{m}\)
- \(\Delta x = 3 \times 10^{-2} \, \text{m}\)

Thus, we have:
\[ d = \frac{(600 \times 10^{-9} \, \text{m}) \times (4 \, \text{m})}{3 \times 10^{-2} \, \text{m}} \]

4. **Calculating \(d\)**:
Performing the calculation:
\[ d = \frac{2400 \times 10^{-9} \, \text{m}}{3 \times 10^{-2}} = \frac{2400 \times 10^{-9}}{0.03} \]
\[ d = 8.0 \times 10^{-5} \, \text{m} = 0.00008 \, \text{m} = 0.08 \, \text{cm} \]

5. **Final Result**:
The separation between the slits is:
\[ d = 0.08 \, \text{cm} \]