w Amicable separationA parallel beam of monochromatic light strikes a transparent prism. The cross section of the prism is a regular hexagon. The beam is parallel to the top and bottom faces of the prism, and points A and B in the diagram are the midpoints of the corresponding edges. After the refraction, two separate parallel beams of light emerge from the prism. What is the minimum index of refraction of the material of the prism that allows such an effect?

Solution: Consider the diagram of the hexagonal prism below, along with the part of the light beam entering between points C and A.

Since the side containing A and C is parallel to the side containing A9 and C9 the beam emerging from the strip A9C9 will be parallel to the beam entering the strip AC. By symmetry, another parallel beam exits from the side just above point A9, satisfying the requirements of the problem. The diagram shown is for the minimum index of refraction that causes this effect. A larger index would move points A9 and C9 down their side, creating a blank region between the two parallel beams.

Physics Challenge for Teachers and Students

Boris Korsunsky, Column EditorWeston High School, Weston, MA 02493 korsunbo@post.harvard.edu

Solutions to May 2012 Challenge

THE PHYSICS TEACHER Vol. 50, 2012

Suppose that the hexagon has side length L. The distance from midpoint A to midpoint C9 is

2 cos 30 3 .L L= This segment also serves as the nor-mal line to the rays entering or exiting the prism at A or C9. The length of the ray passing from A to A9 can be found using the Pythagorean theorem,

( ) ( )222 133 .2L L L+ =

Snells law can now be used to find the minimum index of refraction. 1 . sin 30o = n . sin q ,

1 12 13

,n= .

Finally, .

132 1.803n=

(Contributed by Jason L. Smith, Richland Community College, Decatur, IL)

We would also like to recognize the following contribu-tors: Sharmila Balamurugan (student, Womens Christian

College, Chennai, India)Hratch Barsoumian (Haigazian University, Beirut,

Lebanon)Andr Bellemans (Universit Libre de Bruxelles,

Belgium) Phil Cahill (The SI Organization, Inc., Rosemont PA)David A. Cornell (emeritus, Principia College, Elsah, IL)Julien Desmarets (Lyce Branly, Nogent-sur-Marne,

France)Oscar Escucha Garca (student, Escuela Politcnica Su-

perior, Universidad de Sevilla, Sevilla, Spain) ngel Garca Rivera (student, Escuela Politcnica Supe-

rior, University of Seville, Seville, Spain)Josh Gates (Tatnall School, Greenville, DE)Fredrick P. Gram (Cuyahoga Community College,

Cleveland, OH)

A

B

A

C

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A

C

Norge Cruz Hernndez (Universidad de Sevilla, Sevilla, Spain)

Gerald E. Hite (TAMUG, Galveston, TX)Jos Ignacio iguez de la Torre (Universidad de

Salamanca, Salamanca, Spain)Kenneth E. Jesse (emeritus, Illinois State University,

Normal, IL)Gabriel Kessler (Oldham County High School, Buckner,

KY)Dieter Kuhn (CH-8180 Buelach, Switzerland)Gabriel Lisboa (student, Colgio Integrado Objetivo, So

Paulo, Brazil)Kuan-Fu Liu (student, National Taiwan University, Taipei,

Taiwan R.O.C.) Daniel Mixson (Naval Academy Preparatory School,

Newport, RI)Carl E. Mungan (U.S. Naval Academy, Annapolis,

MD) Thomas Olsen (Society of Physics Students, AIP, College

Park, MD)Eduardo E. Rodrguez (Instituto de Industria, Universidad

Nacional de General Sarmiento, Buenos Aires, Argentina)

Jorge Salazar (Seccin Fsica, Pontificia Universidad Catlica del Per, Per)

Clint Sprott (University of Wisconsin Madison, WI)Ryan Yancey (Lancaster High School, Lancaster, CA)

Many thanks to all contributors and we hope to hear from you in the future!