Young's Double Slit: Find Min. Plexiglas Thickness for Dark Spot

[Feodalherren]

Homework Statement

In a Young’s double-slit experiment using light of wavelength
λ, a thin piece of Plexiglas having index of refraction
n covers one of the slits. If the center point on the
screen is a dark spot instead of a bright spot, what is the
minimum thickness of the Plexiglas?

Homework Equations

The Attempt at a Solution

If the center is to be a dark spot then the Plexiglas must delay the light by 1/2 λ.
Call the distance from each slit to the center d.
Call the thickness of the glass t.

Therefore

$\frac{d}{\lambda} - (\frac{t}{\lambda/n} + \frac{d-t}{\lambda}) = \frac{1}{2}$

My reasoning is that this must be true because there is an extra wavelength in one of the paths. The book, however, reverses it. It takes what I have inside the parenthesis and subtracts that from d/λ. Other than that we agree. Why does the book reverse it?

 

[ehild]

Your equation results in negative t as n is greater than 1. ehild

 

[SammyS]

Homework Statement

In a Young’s double-slit experiment using light of wavelength
λ, a thin piece of Plexiglas having index of refraction
n covers one of the slits. If the center point on the
screen is a dark spot instead of a bright spot, what is the
minimum thickness of the Plexiglas?

Homework Equations

The Attempt at a Solution

If the center is to be a dark spot then the Plexiglas must delay the light by 1/2 λ.
Call the distance from each slit to the center d.
Call the thickness of the glass t.

Therefore,

$\frac{d}{\lambda} - (\frac{t}{\lambda/n} + \frac{d-t}{\lambda}) = \frac{1}{2}$

My reasoning is that this must be true because there is an extra wavelength in one of the paths. The book, however, reverses it. It takes what I have inside the parenthesis and subtracts that from d/λ. Other than that we agree. Why does the book reverse it?

What you mean is not clear.

You are subtracting what's in parentheses from λ/d .

... and ditto to what ehild said.

 

[Feodalherren]

Yes but the book does (stuff) - l/d.
So it does it in reverse. That makes no sense to me. The beam that travels through the plexi should be 1/2 lambda shorter and therefore their equation should equal -(1/2) in my mind.

 

[ehild]

Yes but the book does (stuff) - l/d.
So it does it in reverse. That makes no sense to me. The beam that travels through the plexi should be 1/2 lambda shorter and therefore their equation should equal -(1/2) in my mind.

What is (stuff)-1/d?

I do not think that the book subtracts λ/d. It is d/λ instead is it not?

$(\frac{t}{\lambda/n} + \frac{d-t}{\lambda})-\frac{d}{\lambda} = \frac{1}{2}$

And that is correct.

As the refractive index is higher than 1 in the plexi slab, the phase of the light wave changes more than in air. We say that the optical path difference between the waves should be |λ/2| in order to produce a black central spot. The optical distance is refractive index times physical distance. Both waves travel equal physical distances to the central spot, but the optical distance is nt for the plexi and d-t for air for the ray traveling through the plexiglass, while it is nd for the other ray. The ray though the plexiglass traveled a longer optical distance, its phase changed more than those of the other ray.

Your formula results in negative thickness for the glass slab which is impossible.

ehild

 

[projjal]

The beam that travels through the plexi should be 1/2 lambda shorter and therefore their equation should equal -(1/2) in my mind.

Thats where you made the mistake.The beam passing through the plexi covers larger optical path. So its path would be 1/2λ greater than the other.