Length of the box based on the fringe pattern? I think I'm close.

[spaghed87]

Homework Statement

The figure

shows a neutron in a one-dimensional box. If the right end of the box is opened, the neutron travels out of the box, impinges on a double slit, and is detected 2.0 m behind the double slit. Repeating the experiment over and over produces the fringe pattern shown in the figure.

What is the length (in nm) of the box?

Homework Equations

Variables in equations explained in "The attempt at a solution"

Fringe position:
y(m)=(m*lamda*L)/(d)

lamada(m)=(4*L)/(m)

Fringe spacing:
(delta)y=(lamda*L)/(d)

The Attempt at a Solution

I thought I could use the fact that the fringe position is y(m)=(m*lamda*L)/(d) where lamda is the wavelength, m is the number of fringes which should be two (m=2) since the central max. is m=0. Then L is the length behind the double slit. d is the fringe spacing. I then plugged that into the formula for the wavelength of a open-closed tube which is lamada(m)=(4*L)/(m) where m in this case is m=4 since there are 4 modes in the wave. The two fours cancel out giving lamda=L so, L=0.0165nm but that is not right. So, I tried using the fact that the fringe spacing (instead of fringe position) was (delta)y=(lamda*L)/(d) and I got an answer for L=0.033nm. I know I am close... I'm not the best at physics so can someone spot where I am going wrong?

 

[Redbelly98]

1. I wouldn't use L in the diffraction fringe equation, since it could get confused with the length of the box L.

2. What numbers are you using for Δy and d in the diffraction fringe equation? And what do you get for λ as a result?

3.

lamada(m)=(4*L)/(m)

Please check that equation. I'm pretty sure the "4" should be a "2".

 

[thomate1]

I would approach this problem by first considering the variables and equations involved. The problem states that the fringe pattern is produced by a neutron traveling out of a one-dimensional box, impinging on a double slit, and being detected 2.0 m behind the double slit. The fringe pattern is described by the equation y(m)=(m*lamda*L)/(d), where y is the fringe position, m is the number of fringes, lamda is the wavelength, L is the length behind the double slit, and d is the fringe spacing.

Next, I would consider the other equation given, lamada(m)=(4*L)/(m), which relates the wavelength to the length of the box. This equation is derived from the open-closed tube experiment, but it may not be applicable in this situation since the box is one-dimensional and the experiment is performed with a neutron, not sound waves.

Based on the given information, it seems that we need to find the wavelength in order to determine the length of the box. We can use the fringe spacing equation, (delta)y=(lamda*L)/(d), to solve for the wavelength. However, we need to know the value of d, which is not given in the problem. This is where the number of modes, m, comes into play. The number of modes is equal to the number of slits in the double slit, which in this case is 2. Therefore, we can substitute m=2 into the equation lamada(m)=(4*L)/(m) to get lamda=2L.

Now, we can plug this value into the fringe spacing equation, (delta)y=(lamda*L)/(d), and solve for d. Once we have the value of d, we can use it in the original fringe position equation, y(m)=(m*lamda*L)/(d), to solve for L, the length of the box.

In summary, the length of the box can be determined by finding the wavelength using the number of modes, and then using that value to solve for the fringe spacing and the length of the box. It is important to consider the variables and equations involved in order to come to a correct solution.