In a Young's double-slit experiment, the position of bright and dark fringes on the screen can be determined by using the following formulas:

1. Bright fringes (constructive interference): $y_m = \frac{m \lambda D}{d}$ where:

   • $y_m$ is the distance of the $m$-th bright fringe from the central maximum,
   • $m$ is the fringe order (an integer),
   • $\lambda$ is the wavelength of light,
   • $D$ is the distance between the slit and the screen,
   • $d$ is the distance between the slits.

2. Dark fringes (destructive interference): $y_n = \frac{(n + \frac{1}{2}) \lambda D}{d}$ where:

   • $y_n$ is the distance of the $n$-th dark fringe from the central maximum,
   • $n$ is the fringe order for the dark fringes (an integer).

Given Information:

• Wavelength $\lambda = 0.65 \, \mu m = 6.5 \times 10^{-7} \, m$,
• Distance between the slit and the screen $D = 1.5 \, m$,
• The distance between the middle fringe (central maximum) and the 2nd bright fringe on the negative side ($m = -2$) and the 5th dark fringe on the positive side ($n = 5$) is $51.33 \, mm = 51.33 \times 10^{-3} \, m$.

Let’s break the problem into two steps:

1. Calculate the position of the 2nd bright fringe:

For the 2nd bright fringe, $m = -2$: $y_{\text{bright}} = \frac{-2 \times 6.5 \times 10^{-7} \times 1.5}{d}$

2. Calculate the position of the 5th dark fringe:

For the 5th dark fringe, $n = 5$: $y_{\text{dark}} = \frac{(5 + \frac{1}{2}) \times 6.5 \times 10^{-7} \times 1.5}{d}$

3. The total distance between the fringes is:

$\Delta y = y_{\text{dark}} - y_{\text{bright}} = 51.33 \times 10^{-3} \, m$

We can solve these two equations simultaneously to find the slit separation $d$, and hence, the exact positions of the fringes.

Let me calculate it for you.The distance between the slits, $d$, is approximately $0.142 \, mm$.

Would you like to explore more details, or have any other questions? Here are some related questions:

1. How does the wavelength affect the fringe spacing in a Young's experiment?
2. What happens to the fringe pattern if the slit separation decreases?
3. How do we calculate the position of dark fringes in a single-slit diffraction experiment?
4. Can Young’s double-slit experiment be performed with sound waves?
5. What role does coherence play in the double-slit experiment?

Tip: The fringe spacing increases as the wavelength increases or the slit separation decreases.