A beam of white light traveling in air strikes the face of an equilateral prism

Question

A beam of white light traveling in air strikes the face of an equilateral prism at an incident angle of ?1 = 50° as shown. The glass that makes up the prism has an index of refraction of n = 1.446 for red light and n=1.532 for violet light.

1)What is ?4 as a function of ?2? (A number with units may be used here.)

2)What is ?4 as a function of ?1? (A number with units may be used here.)

3)For red light, what is the expression to calculate ?4? (This is the expression from the previous question with numbers plugged in.)

4)To one decimal place, what is the angle ?4 in degrees for red light?

Please explain and show all your work! Thank you.

Explanation / Answer

1)

Let us assume that

Refractive index for white light n(w)

sin i /sin r = n(w)

sin 50 /sin 2 = n(w)

2 = sin^-1 (0.766044443/n(w)) .............................(1)
-----------------------
From the figure it is seen that the triangle is an equilateral triangle.

Angle of the prism = 60°
Let the normal meet at a point making an angle N

60° + N =180° hence N = 120°
2 + 3 +120= 180°
3 + 120°= 180°
3 = 60°- 2
n(w) = sin 4 / sin 3

n(w) = sin 4/ sin (60°- 2)

4 = sin^-1 (n(w)((3/4)^0.5 cos 2 - sin 2/2)) .........(2)

2)

As we know that

sin 1 /sin 2 = n(w)

So,

2 = sin^-1 (sin1/n(w))

Put the value of 2 in equation (2)

4 = sin^-1 (n(w)((3/4)^0.5 cos 2 - sin 2/2))

4 = sin^-1 (n(w)((3/4)^0.5 cos (sin^-1(sin1/n(w))) - sin ({sin^-1(sin1/n(w) ) }/2 ) ) )

3)

For red light

n = 1.446

Put the value in equation (1) and (2)

2 = sin^-1 (0.766044443/n(r)) = 31.9897769 degree

From the figure it is seen that the triangle is an equilateral triangle.

Angle of the prism = 60°
Let the normal meet at a point making an angle N

60° + N =180° hence N = 120°
2 + 3 +120= 180°
3 + 120°= 180°
3 = 28.0102231°
n(r) = sin 4 / sin 3

1.446 = sin 4/ sin (28.0102231°)

4 = sin^-1 (1.446((3/4)^0.5 cos - sin 2/2)) = 42.77207953 degree

4)

4 = sin^-1 (1.446((3/4)^0.5 cos - sin 2/2)) = 42.77207953 degree

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.