Calculating Number of Fringes with Wavelength and Slit Width

[StudentofPhysics]

1. How many dark fringes will be produced on either side of the central maximum if light ( = 553 nm) is incident on a single slit that is 5.1 10-6 m wide?



2. sin theta = m (lamba/ W)

lamba = wavelength
W = slit width
m - number of fringes



3. I have no idea how to do this without at least an L or theta.

Does anyone know what I am to do?

 

[StudentofPhysics]

ok, i decided to try the slit width divided by the wavelength. This gave me the correct answer.

Is this because this is a way to determine how many wavelengths will fit through the slit and therefore how many fringes there will be?

I just want to make sure I am using a correct theory here and that I did not just get lucky.

 

[m2003]

I can provide you with a step-by-step approach to calculate the number of fringes produced on either side of the central maximum.

1. First, we need to understand the equation given in the content, which is sin theta = m (lambda/W). This equation is known as the grating equation and it relates the angle of diffraction (theta) to the wavelength of light (lambda), the distance between the slits (W), and the order of the fringe (m).

2. In this case, we are given the value of lambda (553 nm) and W (5.1 x 10^-6 m). We can plug in these values in the equation to solve for the angle of diffraction (theta).

sin theta = m (553 x 10^-9 m / 5.1 x 10^-6 m)

theta = sin^-1 (m x 0.108)

3. Now, to calculate the number of fringes, we need to know the value of theta. As mentioned in the content, we would need either the length (L) or the angle (theta) to calculate the number of fringes. If L is not given, we can use the small angle approximation, which states that for small angles, sin theta is approximately equal to theta in radians. Therefore, we can use theta as the number of fringes in this case.

4. Putting all the values together, we get:

Number of fringes = theta = sin^-1 (m x 0.108)

5. To calculate the number of fringes on either side of the central maximum, we need to consider both positive and negative values of m. This is because the grating equation gives us the position of fringes on both sides of the central maximum.

Number of fringes on one side = theta/2 = sin^-1 (m x 0.108)/2

Therefore, for the given values, the number of fringes on one side would be:

Number of fringes on one side = sin^-1 (m x 0.108)/2

= sin^-1 (0.5 x 0.108)

= sin^-1 (0.054)

= 3.1 degrees

Hence, the total number of fringes on both sides of the central maximum would be approximately 6 fringes.

I hope this explanation helps you