Question

the central diffraction max contains exactly seven interference fringes and in this case d/a=4 what must the ratio d/a be if the central max contains exactly five fringes?how many fringes are contained within the first diffraction max on side of the central max? please answere step by step. thanks alot.

Answer 1

The central maximum in a double-slit experiment contains interference fringes based on the ratio of slit separation to slit width (d/a). If seven fringes are seen when d/a=4, the ratio for five fringes would be d/a = 20/7. Each side of the central maximum would contain half the odd number of fringes, excluding the central maximum itself.

Step-by-step explanation:

The question deals with the topic of diffraction and interference in physics, particularly within double-slit experiments. The central diffraction max (or central maximum) consists of the brightest fringe in the middle of the pattern, with several lighter fringes on either side if the experiment setup allows for them. The number of fringes within the central maximum depends on the ratio of the slit separation (d) to slit width (a).

Given that seven fringes are observed within the central maximum when d/a = 4, to find the new ratio for which exactly five fringes are observed, we recognize that the number of fringes (n) is proportional to the width of the central maximum, which in turn is directly proportional to the ratio d/a. Thus, if n is reduced from seven to five, which is a reduction to 5/7 of the original count, the ratio d/a must also be reduced to 5/7 of its original value. This implies the new d/a ratio would be:

d/a = 4 × (5/7) = 20/7

It's also important to note that the number of fringes to one side of the central max in the first diffraction maximum would be half the odd number of fringes in the central maximum, because the pattern is symmetrical. Therefore, if the central max contains five fringes, then there would be two fringes within the first diffraction maximum on one side of the central maximum (excluding the central maximum itself).

Answer 2

To find the ratio of d/a resulting in the central max containing 5 fringes, use the formula n = (2Dλ) / d. Substitute the values and solve for D. The number of fringes in the first diffraction max can be found using N = (Dλ) / (a × d).

Step-by-step explanation:

To find the ratio of d/a that would result in the central max containing exactly five fringes, we can use the formula:

n = (2Dλ) / d

Where n is the number of fringes, D is the distance between the screen and the slit, λ is the wavelength of light, and d is the slit width.

Since we know that n = 7 when d/a = 4, we can substitute the values and solve for D.

7 = (2 ×D ×λ) / (4 ×a)

Similarly, when n = 5, we can rearrange the formula to solve for d/a:

d/a = (2 ×D ×λ) / (5 ×n)

Once we have the value of d/a, we can calculate the number of fringes within the first diffraction max on one side of the central max using the formula:

N = (Dλ) / (a ×d)

Where N is the number of fringes.