Problem: Plastic Covering One Slit in Double Slit Experiment

[chipwhisperer]

Homework Statement

A very thin sheet of plastic (n = 1.45) covers one slit of a double-slit apparatus illuminated by 570 nm light. The center point on the screen, instead of being a maximum, is dark. What is the (minimum) thickness of the plastic?

Homework Equations

Constructive Interference:
d *sin(theta)=m*lamda

Destructive Interference:
d *sin(theta)=(m+1/2)*lamda

lamda_n=lamda/n

The Attempt at a Solution

So I'm a TA for a physics class and I'm a little stumped on explaining this problem. The solutions manual says that the phase shift must be an odd multiple of one half. I got that. However, it then proceeds to write down this equation:

N= t/lamda_n - t/lamda = 1/2

where t is the thickness of the plastic. Any idea how they got this relation? I would really appreciate an explanation.

Thanks!

 

[stunner5000pt]

t = N lambda right.
What does N represent?

number of what??

 

[chipwhisperer]

N equals the phase shift I believe, i.e. 1/2, I'm not exactly sure though

 

[Doc Al]

So I'm a TA for a physics class and I'm a little stumped on explaining this problem. The solutions manual says that the phase shift must be an odd multiple of one half. I got that. However, it then proceeds to write down this equation:

N= t/lamda_n - t/lamda = 1/2

where t is the thickness of the plastic. Any idea how they got this relation? I would really appreciate an explanation.

Well, that expression is the phase difference between the two beams that converge at the center, expressed in terms of wavelengths. (lambda is the wavelength of light in air (or vacuum), where n = 1; lambda_n is the wavelength of light in the plastic, which is lambda/n.) The physical path lengths are the same; the only difference is that one passes through a thickness of plastic (phase shift = t/lamda_n) while the other beam passes through the same thickness of air (phase shift = t/lambda).

 

[chipwhisperer]

Okay, I get that. But, why is the phase shift = t/lamda? I just don't see that direct correspondence.

 

[Doc Al]

But, why is the phase shift = t/lamda?

If a lightwave travels from points A to B, do you not agree that the phase of the light at B differs from the phase it had at A? And that the phase shift in moving from A to B can be expressed by the number of wavelengths contained in the distance between A and B? For example, if the distance A-to-B is 1/2 lambda, when the light reaches B it is exactly 180 degrees out of phase compared to its phase when at A?

Let me know if this makes sense so far.