A horizontal incident beam consisting of white light passes through an equilater
ID: 2203832 • Letter: A
Question
A horizontal incident beam consisting of white light passes through an equilateral prism, like the one shown in the figure. What is the dispersion (?v - ?r) of the outgoing beam if the prism's index of refraction is nv = 1.505 for violet light and nr = 1.455 for red light?Explanation / Answer
9.2° ... Equilateral triangle —> ? ( 180° ) = 60° interior angles … since, on one hand, the incident ray along the horizontal is perpendicular to the bisector of the top interior angle of the equilateral triangle, and on the other hand, the line N1 N1' normal to the entry point is perpendicular to the left side of the triangle, then the angle of incidence ?1 with respect to N1 N1' at the entry point on the left side of the prism is equal to half of the top interior angle of the equilateral triangle … ?1 = ½ ( 60° ) = 30° … so that for the first refraction of violet light ( n = 1.485 ) from air to glass … … ( 1.000 ) sin 30° = ( 1.485 ) sin ?2 …-->… sin ?2 = ( 1.000 / 1.485 ) sin 30° … ?2 = sin ? ¹ [ ( 1.000 / 1.485 ) sin 30° ] = 19.6760° (about 20°) … Now, the hypothetical, undeflected, outgoing violet light also makes an angle of around 20° with respect to the line (call it L1) at the exit point, going back to the air again, that is parallel to the normal line N1N1' at the left side of the prism … … since the horizontal line (call it L2) at the exit point and line L1 makes an angle equal to the incident angle of 30° during the first refraction from air to glass, then the normal line N2 N2' at the exit point and the hypothetical, undeflected, outgoing violet light makes an angle of 40.324°, which is equal to the angle of incidence during the second refraction, when the refracted violet light goes back to air again … therefore, for the second refraction of violet light going from glass to air … n2 sin ?2 = n3 sin ?32 … n2 = 1.485 … ?2 = 40.324° … n3 = 1.000 … … sin ?32 = ( n2 / n3 ) sin ?2 = n2 sin ?2 = ( 1.485 ) sin 40.324° … ?32 = sin ? ¹ [ ( 1.485 ) sin 40.324° ] = 73.94° … that is, the outgoing violet light at the right side of the prism makes an angle of 73.94° with the normal … Following the same analysis described above for violet light, we get the following for red light ( n = 1.425 ) … for the first refraction from air to glass … … ( 1.000 ) sin 30° = ( 1.425 ) sin ?21 …-->… sin ?21 = ( 1.000 / 1.425 ) sin 30° … ?21 = sin ? ¹ [ ( 1.000 / 1.425 ) sin 30° ] = 20.5410° … For the second refraction of red light from glass to air … … n2 sin ?2 = n3 sin ?31 … n2 = 1.425 … ?2 = 39.459° … n3 = 1.000 … … sin ?31 = ( n2 / n3 ) sin ?2 = n2 sin ?22 = ( 1.425 ) sin 39.459° … ?31 = sin ? ¹ [ ( 1.425 ) sin 39.459° ] = 64.71° … that is, the outgoing red light at the right side of the prism makes an angle of 64.71° with the normal … Finally ... the dispersion ?v – ?r = 73.94° - 64.71° = 9.2°