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Helping the Moon take a selfie

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2014 Phys. Educ. 49 486

(http://iopscience.iop.org/0031-9120/49/5/486)

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Physics Education 49 (5) 486

W H Baird

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486

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© 2014 IOP Publishing Ltd

2014

49

Phys. educ.

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0031-9120

10.1088/0031-9120/49/5/486

Physics education

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F r o n t l i n e s

IntroductionImaging using the plane (flat) mirror is a common starting point for introducing the laws of optics. The law of reflection specifies that the angles of incidence and reflection (both measured from the normal to the mirror) must be equal:

θ θ=i r (1)

This is enough to show that a mirror need only be half of one’s height to enable viewing of one’s entire body (a problem mentioned in many introductory texts [1–5]). It is easier to illustrate this if we assume the eyes are at the top of the head (figure 1).

While no particular distance between the person and mirror is specified, some years ago a student asked an interesting question: assume you have a small makeup or compact mirror and you use it to look over your shoulder at the Moon, which fits comfortably inside the compact’s field of view (figure 2). If someone on the Moon has a huge telescope, large enough to enable that per-son to easily see you and your mirror, can they also see the entire Moon in the compact, just as you do?

This question served as a kind of turning point for me, as it was asked very early on in my

teaching career. In the very beginning, I used to be afraid that a student would ask a question I couldn’t answer; now, when this happens, there is always some excitement that I am going to learn something new about basic physics!

I have found this to be a thought-provoking question when I have since asked it of my intro-ductory physics students. Rather than just giving them the answer, even after some discussion, I tell them to solve it experimentally. They can’t build a telescope on the Moon, of course, but they can back away from a small mirror, increasing their distance and using binoculars to maintain their view. When they try this, they find that the mirror contains the same fraction of their body whether near or far (figure 3).

The easiest explanation I have found is to encourage the students to look at the angular sizes of the objects involved. When the compact mirror is held at arm’s length, we can assume a diameter of 10 cm and a distance of 100 cm. The small angle equation  says that the mirror will subtend an angle of 1/10 radian, or about 5.7°. The Moon’s diameter is 3476 km and its mean distance from Earth is 384 400 km, for an angular size of just over 0.5°, easily fitting inside our mirror.

Helping the Moon take a selfieWilliam H Baird

Department of Chemistry & Physics, Armstrong State University, Savannah, GA 31419, USA

E-mail: william.baird@armstrong.edu

AbstractIt is a fundamental result of introductory optics that a plane mirror must be at least one half of your height if you want to see your entire body. Students are commonly confused about whether this is still true as you back very far away from the mirror. An interesting student question proposed that we observe the Moon’s image in a small makeup mirror. If someone on the Moon had a telescope large enough to see you and your surroundings clearly, would that person also be able to peer over your shoulder and see the entire Moon in your mirror, as you do? The answer provides a useful ‘view’ on mirror reflections.

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0031-9120/14/050486+3$33.00 © 2014 IOP Publishing Ltd

Helping the Moon take a selfie

487Physics EducationSeptember 2014

When the person on the Moon looks down at the compact mirror, its angular size will be approximately 2.6  ×  10−10 radians. The Moon’s image (in a suitable mirror) would have a size equal to its diameter divided by twice the Earth–Moon distance, or about 4.5  ×  10−3 radians. For the Moon occupant, the Moon’s image is some 17.3 million times the angular size of the mir-ror. A quick check reveals that the mirror size required is then 17.3 million times 10 cm, or 1730 km. This is, unsurprisingly, half the size of the Moon.

Imagining the mirror as a kind of peephole onto the virtual images inside makes it a little clearer. If your eye is only 1 cm from a peep-hole (even one without a lens) you can see most or all of someone on the other side of the door. If you were to back up until you were each 1 m from the door and then try to see through the peephole, you would see only a tiny fraction of the person. The Moon’s angular size is smaller than the mirror’s from our viewpoint, but from the Moon, its virtual image is still far larger than the mirror in our hands.

Figure 1. A plane mirror showing the path of light rays reflecting from the feet (dashed line) as well as the top of the head (dotted line). If the mirror is one half the person’s height, the image of the head will result from rays reflected from the top of the mirror, while the image of the feet is due to rays reflected from the bottom of the mirror.

Figure 2. The Moon as seen through a 12.5 cm diameter octagonal mirror.

W H Baird

488 Physics Education September 2014

AcknowledgementsThe author thanks Faye Montgomery for the pho-tographs in figures  2 and 3 and the anonymous referee for helpful suggestions.

Received 22 April 2014, in final form 1 May 2014,accepted for publication 20 May 2014doi:10.1088/0031-9120/49/5/486

References[1] Serway R A and Jewett J W 2006 Principles of

Physics: A Calculus-Based Text 4th edn (Belmont, CA: Brooks/Cole–Thomson Learning) 891

[2] Tipler P A and Mosca G 2008 Physics for Scientists and Engineers Extended Version 6th edn (New York, NY: W H Freeman and Company) 1135

[3] Cutnell J D and Johnson K W 2009 Physics 8th edn (Hoboken, NJ: John Wiley and Sons) 777

[4] Knight R D 2012 Physics for Scientists and Engineers: A Strategic Approach with Modern Physics 3rd edn (Boston, MA: Addison- Wesley) 660

[5] Etkina E, Gentile M and Van Heuvelen A 2013 College Physics 1st edn (Boston, MA: Addison-Wesley) 812

Figure 3. The same fraction of the photographer’s face is visible whether near (left) or far (right) from the mirror. This experiment can also be performed with cheap binoculars. Notice that the edge of the mirror is not in focus due to the fact that it is twice as close to the camera as the photographer’s image.

Bill Baird is an associate professor of physics at Armstrong State University, Savannah, GA, USA, where he has taught for nine years. His primary research interest is the design, construction and use of sensors.