Two parallel slits are illuminated by light composed of two wavelengths, one of which is 645 nm. On a viewing screen, the light whose wavelength is known produces its fifth dark fringe at the same place where the light whose wavelength is unknown produces its fifth-order bright fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength?

Question

Two parallel slits are illuminated by light composed of two wavelengths, one of which is 645 nm. On a viewing screen, the light whose wavelength is known produces its fifth dark fringe at the same place where the light whose wavelength is unknown produces its fifth-order bright fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength?

___________ nm

Answer 1

that did not work.

Answer 2

To solve this problem, we'll use the equation for the location of the bright fringes of a double-slit interference pattern:

y = (m * λ * L) / d

where:
y is the distance on the screen from the central bright fringe to the m-th bright fringe (measured perpendicular to the line connecting the slits),
m is the order of the bright fringe,
λ is the wavelength of light,
L is the distance from the slits to the screen,
d is the distance between the two slits.

We are given that the known wavelength (λ1) produces its fifth dark fringe at the same location as the unknown wavelength (λ2) produces its fifth bright fringe.

For the known wavelength:
m1 = 5
y1 = y2 (since the fringes are at the same location)
λ1 = 645 nm

For the unknown wavelength:
m2 = 5
y2 = y1 (since the fringes are at the same location)
λ2 = ?

We can equate the two equations for the y-values:

(m1 * λ1 * L) / d = (m2 * λ2 * L) / d

Substituting the known values:

(5 * 645 nm * L) / d = (5 * λ2 * L) / d

Simplifying:

λ2 = (645 nm * d) / L

To find the unknown wavelength (λ2), we need to know the values of d and L. These values are not provided in the question, so we cannot determine the exact unknown wavelength without this information.

Answer 3

To find the unknown wavelength, we need to use the concept of fringe spacing in interference patterns. In a double-slit interference pattern, the fringe spacing (distance between adjacent fringes) can be determined by the equation:

d * sin(theta) = m * lambda

where:
- d is the separation between the slits
- theta is the angle between the central bright fringe and the mth-order fringe
- m is the order of the fringe
- lambda is the wavelength of the light

In this problem, we are given that the known wavelength (645 nm) produces its fifth dark fringe at the same place where the unknown wavelength produces its fifth-order bright fringe. This means that the path difference between these fringes is an integer multiple of the wavelength, which results in constructive interference.

For the known wavelength (645 nm), the path difference between the fifth dark fringe and the zeroth-order bright fringe is 5 * lambda.

For the unknown wavelength, the path difference between the fifth-order bright fringe and the zeroth-order bright fringe is also 5 * lambda.

Since these two path differences are equal, we can set them equal to each other and solve for the unknown wavelength:

5 * (645 nm) = 5 * lambda

Now, we can solve for lambda:

lambda = (5 * 645 nm) / 5

lambda = 645 nm

Therefore, the unknown wavelength is also 645 nm. Hence, the answer is 645 nm.

Answer 4

n*Lambda= xd/L

Now in this case, xd/l is a constant, so

n1*Lambda1= (n2-1/2)lambda2 if I follow the numbers right, then n1 is 5, lambda1 is 645nm, n2 is 5, and lambda2 is the unknown.

Notice in counting, the first dark fringe occurs before the first light fringe. So at the fifth light fringe, the fifth dark fringe is just before it.That is why I put the -1/2. Check my thinking on that, the wording has me a little bothered.