Light, optics and radians

[arc5005]

Maybe I have forgotten everything I know about radians, so I'm lost in regards to this question:

Is radians something I should briefly review before the MCAT?

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PASSAGE: IMG 5932 < - picture of the passage

QUESTION:
If in using Set-up 2, the phase difference is measured to be π radians, then which of the following is a possible value for the thickness of the glass, if the incident angle is assumed to be negligibly small?

ANSWERS:
A) 0.250 m
B) 0.375 m
C) 0.500 m
D) 0.625 m


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Answer:

A) 0.250 m

If the light ray is coming in approximately perpendicular to the surface of the glass, then delta r = 2t, where t represents the thickness of the glass, because the light ray that reflected off of the second surface must travel into the glass and back out. The question states that the waves are π out of phase, so this problem becomes a simple calculation:

delta r = 2t = ( lambda / 2 pi)Phi & t = ( lambda / 4 pi)Phi = (1m / 4 pi rad)(pi rad) = 0.25 m



I'm not following the math here. How did they decide that lambda = 1 m, and does Phi = pi radians = phase difference????? (the phase difference is measured to be pi radians)?

Thank you in advance.

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[AdaptPrep]

Hi, arc5005.

The second to the last sentence says that the wavelength of the radio wave used in this experiment was 1.0 meters (I almost missed it too). Phase difference can be measured in degrees or radians (where 360 degrees is 2pi radians; 57.3 degrees = 1 radian). In this problem, that really was not necessary to know, but it may come up in other problems. So you are correct when you said the phase difference is measured to be pi radians.

In this problem, the phase difference, phi, was 1pi radian and would cancel:
delta r = 2t
2t = (lambda/2pi) phi = (lambda/2pi) pi = lambda/2
t = lambda/4
Now substitute 1 meter for lamba:
t = 1/4 = 0.250 meters

I hope that helps.

Adaptprep

 

[arc5005]

Hi, arc5005.

The second to the last sentence says that the wavelength of the radio wave used in this experiment was 1.0 meters (I almost missed it too). Phase difference can be measured in degrees or radians (where 360 degrees is 2pi radians; 57.3 degrees = 1 radian). In this problem, that really was not necessary to know, but it may come up in other problems. So you are correct when you said the phase difference is measured to be pi radians.

In this problem, the phase difference, phi, was 1pi radian and would cancel:
delta r = 2t
2t = (lambda/2pi) phi = (lambda/2pi) pi = lambda/2
t = lambda/4
Now substitute 1 meter for lamba:
t = 1/4 = 0.250 meters

I hope that helps.

Adaptprep

Yes, this helps a ton, except I am still confused at one point, if you don't mind clarifying please

I understand that Phi = phase difference = 1 pi radians

However, how does the equation go from: ( lambda / 4 pi)Phi

to: (1m / 4 pi rad)(pi rad)

How did a radian get into the denominator? Once it's there, I understand the the algebra, but I'm confused how a radian can be inserted into the denominator when prior it was only 4pi.

Thank you.

 

[AdaptPrep]

Sure! I will try. It might help if you just think of "rad" as units. Just as lambda becomes 1 meter, 4 pi becomes 4 pi rad. It would have helped if the passage clarified and said the units were radians (but they most likely assumed that since there is no degree symbol). 2pi is a pretty good indicator that you are dealing with radians and not degrees. Therefore, it becomes 2pi radians if you use units.

They could have said 2t = (lambda/360 degrees) phi. Then, phi would have been given as 180 degrees (and not 1pi radians):
2t = (1 meter / 360 degrees) 180 degrees = 1 meter / 2
t = 1 meter / 4

I hope that helps.

 

[arc5005]

Sure! I will try. It might help if you just think of "rad" as units. Just as lambda becomes 1 meter, 4 pi becomes 4 pi rad. It would have helped if the passage clarified and said the units were radians (but they most likely assumed that since there is no degree symbol). 2pi is a pretty good indicator that you are dealing with radians and not degrees. Therefore, it becomes 2pi radians if you use units.

They could have said 2t = (lambda/360 degrees) phi. Then, phi would have been given as 180 degrees (and not 1pi radians):
2t = (1 meter / 360 degrees) 180 degrees = 1 meter / 2
t = 1 meter / 4

I hope that helps.

thank you!