Could my textbook sample problem be wrong?

[p3t3r1]

Well, here is the problem..
In summer months, the amount of solar energy entering a house should be minimized. Window glass is made energy efficient by applying a coating to maximize reflected light. Light in the midrange of the visible spectrum (at 568 nm) travels into energy efficient window glass. What thickness of the added coating is needed to maximize reflected light and thus minimize transmitted light?

(air, n = 1.0, coating, n= 1.4 , glass n = 1.5)

Book's solution:
Reflection occurs both at the air-coating interface and at the coating-glass interface. In both cases, the reflected light is 180 deg out of phase with the incident light, since both reflections occur at a fast-to-slow boundary. The two reflected rays would therefore be in phase if there were a zero path difference. To produce constructive intereference the path difference must be wavelength /2. In otherwords, the coating thickness t must be wavelength /4 , where wavelength is the wavelength of the light in the coating.

My solution:
To produce constructive intereference the path difference must be 1 wavelength, not a half wavelength. Therefore, the coating thickness, t, must be wavelength /2. This makes sense since for reflection, we have phase change of ray 1 been reflected and inverted, which is same as change in wavelength /2. We also have ray 2 been reflected and inverted, which is same as change in wavelength /2. Ray 2 also travels more distance, (t x 2, which would be one wavelength if t = wavelength /2) Therefore, ray 2 would differ from ray 1 by a path difference of 1 wavelength and they would interfere constructively for max reflection. And for transmission, we have ray 1 transmitted through without reflection, hence without a phase change caused by the inversion. Then we have ray 2 reflected, undergo a phase change of 180 degrees, or a wavelength/2. It would also travel 1 wave length more since the extra distance is t x 2. Therefore, ray 2 would be half a wavelength from ray 1 and they would interfere destructively to minimize.

The problem is though. if this sample problem is wrong.. so would all the exercises and pratice problem answers.. which would really suck.

 

[ice109]

how does interference affect reflection? are you saying that if the waves were out of phase then the film wouldn't reflect?

 

[Doc Al]

The book's solution is wrong; yours is correct.

Book's solution:
Reflection occurs both at the air-coating interface and at the coating-glass interface. In both cases, the reflected light is 180 deg out of phase with the incident light, since both reflections occur at a fast-to-slow boundary. The two reflected rays would therefore be in phase if there were a zero path difference.

So far, so good.

To produce constructive intereference the path difference must be wavelength /2.

Nonsense. For constructive interference--and maximum reflection--the optical path length difference must be an integer multiple of full wavelengths.

In otherwords, the coating thickness t must be wavelength /4 , where wavelength is the wavelength of the light in the coating.

That would produce an anti-reflection coating.

The problem is though. if this sample problem is wrong.. so would all the exercises and pratice problem answers.. which would really suck.

Check and see for yourself; maybe it's just a fluke. (Inexcusable in a sample problem.) What book are you using?

For a correct discussion of thin film interference, go here: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/thinfilm.html"

 

[p3t3r1]

I am using the standard Ontario Grade 12 Physics textbook. Nelson 12 Physics.

http://www.school.nelson.com/secondary/science/0176259880/default.html

Luckily I have readily access to university textbooks by Sears, Haliday etc.. so It didn't create as a big fuss as it would.

I checked the erreta for the book and there was no mention of this. They could have least check the sample problems. I don't mind having the answers for the pratice answers wrong. At least we could assume their answers are wrong. Having sample problems wrong would misled thousands of students.

 

[Doc Al]

Make sure your instructor is aware of the mistake and reports it to the publisher.

 

[Chris Hillman]

Picking up a theme, textbooks published by reputable publishing houses are generally sent out for checking by independent parties (typically a professor who farms out the task to a graduate student or two). This is an invaluable procedure since it can be very difficult for an author to catch mind blips (oddly enough, the problem is often that he knows too much, not too little). Unfortunately, sometimes mistakes slip through the cracks. This is less common, as I hope and believe, at the university level, but by a regrettable tradition, lower level science textbooks are all to often full of misinformation.

Now compare Wikipedia, where fact checking is chaotic and generally not performed by individuals who have mastered the subject at hand.

 

[ice109]

Picking up a theme, textbooks published by reputable publishing houses are generally sent out for checking by independent parties (typically a professor who farms out the task to a graduate student or two). This is an invaluable procedure since it can be very difficult for an author to catch mind blips (oddly enough, the problem is often that he knows too much, not too little). Unfortunately, sometimes mistakes slip through the cracks. This is less common, as I hope and believe, at the university level, but by a regrettable tradition, lower level science textbooks are all to often full of misinformation.

Now compare Wikipedia, where fact checking is chaotic and generally not performed by individuals who have mastered the subject at hand.

my intro E&M lab book was rife with mistakes