Predicting a decrease in fringe distance (equations)

[chef99]

Homework Statement

a) Imagine that you are conducting an activity with a laser to create an interference pattern. Use the appropriate equations to predict two ways (other than the way described in the following example) to change the interference pattern in order to have closer fringes. Explain your predictions.

example

According to the equation∆x = Lλ / d,the distance between the fringes (∆x) is proportional to the wavelength (since they are both numerators). This means that increasing one will increase the other. Therefore, decreasing the wavelength will decrease the distance between the fringes.

Homework Equations

|P_mS₁ - P_mS₂| = mλ

∆x = Lλ / d

The Attempt at a Solution

1. According to the equation

|P_mS₁ - P_mS₂| = mλ

Increasing the path length, (P_mS₁ and P_mS₂) will decrease the distance between the fringes, as the further away from the screen the light source is, the smaller the fringes become, thus having "closer fringes".2. According to the equation

∆x = Lλ / d

Increasing the slit separation (d) will decrease the distance between the fringes (∆x), because (d) is the denominator, which means increasing its value will decrease the fraction value, and since (∆x) is on the other side of the equation, and the equation must be equal, decreasing the value of one side will decrease the value of the other.

I believe the first answer is correct, but I'm not certain. I'm more confident in the second answer, but again want to make sure I described it properly. Any help on this would be most appreciated.

 

[Gene Naden]

Homework Statement

a) Imagine that you are conducting an activity with a laser to create an interference pattern. Use the appropriate equations to predict two ways (other than the way described in the following example) to change the interference pattern in order to have closer fringes. Explain your predictions.

example

According to the equation∆x = Lλ / d,the distance between the fringes (∆x) is proportional to the wavelength (since they are both numerators). This means that increasing one will increase the other. Therefore, decreasing the wavelength will decrease the distance between the fringes.

Homework Equations

|P_mS₁ - P_mS₂| = mλ

∆x = Lλ / d

The Attempt at a Solution

1. According to the equation

|P_mS₁ - P_mS₂| = mλ

Increasing the path length, (P_mS₁ and P_mS₂) will decrease the distance between the fringes, as the further away from the screen the light source is, the smaller the fringes become, thus having "closer fringes".2. According to the equation

∆x = Lλ / d

Increasing the slit separation (d) will decrease the distance between the fringes (∆x), because (d) is the denominator, which means increasing its value will decrease the fraction value, and since (∆x) is on the other side of the equation, and the equation must be equal, decreasing the value of one side will decrease the value of the other.

I believe the first answer is correct, but I'm not certain. I'm more confident in the second answer, but again want to make sure I described it properly. Any help on this would be most appreciated.

I think your first answer may be incorrect. The angle between the fringes remains the same as you move the screen out. So the distance between the fringes will increase. I am not sure what you mean by "path length." If you increase the distance between the light source and the slits, this might make the fringes sharper - I am not sure - but it will not make them closer together.

Do you have access to a laboratory with a laser? That would be the best way to work this problem - by experiment!

 

[chef99]

I think your first answer may be incorrect. The angle between the fringes remains the same as you move the screen out. So the distance between the fringes will increase. I am not sure what you mean by "path length." If you increase the distance between the light source and the slits, this might make the fringes sharper - I am not sure - but it will not make them closer together.

Do you have access to a laboratory with a laser? That would be the best way to work this problem - by experiment!

Unfortunately, I do not have access to a lab. I do however have a simulation that was provided with my course materials. I can see that moving the slits further from the screen decreases the distance between the fringes, and this equation backs this up. I think perhaps I explain better below;

In to the equation ∆x = Lλ / d,

Decreasing the distance from the slits to the screen (L), will decrease the distance between the fringes (∆x), as L is a numerator of the fraction, meaning if L is decreased, the value of the fraction decreases, and since ∆x is on the opposite side of the equation, and both sides must be equal, then decreasing L will decrease ∆x.

The reason I didn't use this equation originally is that I already used it for my other example, and I think it would be better if I used different equations for the two examples. I'm just not sure how to identify L in the equation |PmS1 - PmS2| = mλ.

 

[kuruman]

The reason I didn't use this equation originally is that I already used it for my other example, and I think it would be better if I used different equations for the two examples.

It's OK to use the same equation. It looks like this is an "equation interpretation" question that is aimed at helping you understand what the equation is saying to you and what the symbols stand for.

 

[chef99]

It's OK to use the same equation. It looks like this is an "equation interpretation" question that is aimed at helping you understand what the equation is saying to you and what the symbols stand for.

Ok, and my explanation is sound?

 

[Gene Naden]

In the equation |PmS1 - PmS2| = mλ, PmS1 is probably the distance from slit 1 to the point on the screen where a fringe is and PmS2 would be the distance from slit 2 to the same fringe :)

 

[kuruman]

Ok, and my explanation is sound?

Yes.