How Do You Calculate Slit Separation in a Double Slit Experiment?

[maccyjj]

Homework Statement

Red plane waves from a ruby laser (694.3nm) in air pass through two parallel slits in an opaque screen. A fringe pattern forms on a distant wall and we see the fourth bright band 1^o above the central axis. (Counting the central bright fringe as zero). Calculate the separation of the slits.

Homework Equations

Phase difference $$\phi$$₂ -$$\phi$$₁ = (2$$\pi$$dsin($$\theta$$))/$$\lambda$$

Amplitude A²_Tot = 2A²(1 + cos($$\phi$$₂ -$$\phi$$₁))

The Attempt at a Solution

I've figured out that when $$\theta$$=0 the amplitude is equal to 4A². I then tried to use this amplitude to find the other maxima (bright bands) but ended up only getting multiples of 2$$\pi$$. I think what I'm lacking is knowing the amplitude definition for a bright band. Could someone help?

 

[mark726]

Thank you for your question. It seems like you are on the right track with your calculations so far. To find the separation of the slits, we need to use the equation for the phase difference between the two waves passing through the slits. This is given by:

\phi2 - \phi1 = (2\pi d \sin(\theta))/\lambda

Where d is the separation of the slits, \theta is the angle between the central axis and the bright band, and \lambda is the wavelength of the laser.

To find the amplitude of the bright bands, we can use the equation you have provided:

A2Tot = 2A2(1 + \cos(\phi2 - \phi1))

Where A2 is the amplitude of each individual wave passing through the slits. As you have correctly noted, when \theta = 0, the amplitude is equal to 4A2. This is because at the central axis, the waves are in phase and their amplitudes add together. However, for the other bright bands, the waves are out of phase and their amplitudes will vary depending on the value of \theta.

To find the amplitude of the fourth bright band, we can use the equation above and plug in the value of \phi2 - \phi1 at that point. We know that the phase difference at the fourth bright band is equal to 4\pi, so we can write:

A2Tot = 2A2(1 + \cos(4\pi))

Now, we know that \cos(4\pi) = 1, so we can simplify the equation to:

A2Tot = 2A2(1 + 1) = 4A2

This means that the amplitude of the fourth bright band is also equal to 4A2, just like at the central axis. From this, we can see that the separation of the slits must be such that the waves passing through them are in phase at the fourth bright band. This will occur when the phase difference is equal to an integer multiple of 2\pi, which in this case is 4\pi. Therefore, we can write:

(2\pi d \sin(\theta))/\lambda = 4\pi

Solving for d, we get:

d = 2\lambda/\sin(\theta)

Now, all we need to do is plug