Light: interference - two slit

• sightempest
In summary, the greatest value of Xp at which the detected light is minimum due to destructive interference is 7.88 x 10^-6 m.
sightempest

Homework Statement

Two isotropic point sources of light (s1 and s2) are separated by 2.7micrometers along a y-axis and emit in phase at wavelength 900 nm and at the same amplitude. A point detector is located at point P at coordinate Xp on the x-axis (s1 is also on the x axis, at (0,0).) What is the greatest value of Xp at which the detected light is minimum due to destructive interference?

I don't know how to make a diagram for this on the computer but say there is an x,y plane: S1 is on the origin, P is at distance Xp to the right of S1 (on the X axis) and S2 is at 2.7um from S2 (lower, on the Y axis.)

Homework Equations

Y = (λL (m+½)) / d
Where Y is the position of the interference minima
L is the distance Xp
D is the slit space (2.7 um)
and λ is the wavelength.
m= mode.

1.$d\sin\theta=m\lambda$ for constructive interference
2.$d\sin\theta=(m+\frac{1}{2})\lambda$ for destructive interference
3.$\phi=\frac{2\pi}{\lambda}(r_2-r_1)$ phase difference related to path difference

The Attempt at a Solution

I figure since the center of interference is always between the two slits (or in this case point sources), y = 1/2 of 2.7um, i.e. 1.35 micrometers, and thus

Y = (λL (m+½)) / d
1.35 x 10 ^- 6 = [ (900 * 10^-9) * L (m+½) / 2.7 * 10 ^ 0.6 ]
or
4.05 x 10 ^ -6 = L (m+½).

And I'm stuck there, I don't know if I'm to suppose that m is 0 seeing as how that would give the biggest L.

The answer is 7.88 x 10^-6 m, but I can't get it...

Last edited:
Rather than try to force fit the formula for two-slit interference to your problem, why not attack the problem directly? Imagine your detector on the x-axis, starting at the origin and moving outward. Now consider the path length difference between the light reaching the detector from both sources. As you move the dectector further along the x-axis, does the path length difference--and thus the phase difference--increase or decrease? What's the minimum path length difference that will lead to destructive interference? Write an expression for the path difference as a function of x, and use the insights from answering the above questions to solve for the maximum value of x.

Thank you, kind sir!

Last edited:

1. What is the two-slit interference pattern?

The two-slit interference pattern is a phenomenon in which light waves passing through two closely spaced slits create a pattern of alternating bright and dark fringes on a screen. This pattern is a result of the interference of light waves from the two slits.

2. How does the distance between the slits affect the interference pattern?

The distance between the slits, also known as the slit separation, is directly proportional to the spacing of the fringes in the interference pattern. As the slit separation decreases, the fringes become closer together, and as the slit separation increases, the fringes become more spread out.

3. What happens when the distance between the slits is equal to the wavelength of light?

When the distance between the slits is equal to the wavelength of light, constructive interference occurs, and a bright fringe is produced. This is because the peaks and troughs of the light waves from each slit align, creating a larger amplitude and a brighter spot on the screen.

4. Can two different colors of light create an interference pattern?

Yes, two different colors of light can create an interference pattern. However, the spacing of the fringes will be different for each color due to their different wavelengths. This is known as chromatic dispersion.

5. What is the difference between single-slit and two-slit interference patterns?

A single-slit interference pattern is created by light passing through a single narrow slit, producing a pattern of bright and dark fringes. In contrast, a two-slit interference pattern is created by light passing through two closely spaced slits, resulting in a more complex pattern of fringes with varying intensities.

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