In the figure on the left, a beam of white light is incident at angle ? = 46° on

Question

In the figure on the left, a beam of white light is incident at angle ? = 46° on a common window pane (shown in cross section). For the pane's type of glass, the index of refraction for visible light ranges from 1.525 at the blue end of the spectrum to 1.510 at the red end. The two sides of the pane are parallel. (Hint: When you look at an object through a window pane, are the colors in the light from the object dispersed as shown in the figure on the right, where chromatic dispersion occurs at the first surface and is increased at the second surface?) Please Show work.

(a) What is the angular spread of the colors in the beam when the light enters the pane?
°

(b) What is the angular spread of the colors in the beam when it emerges from the opposite side?
°

Explanation / Answer

part a:

for the pane:

consider an index of refraction n.

at the first surface where refraction occurs:

incident angle=theta

let refraction angle be theta1.

using snell’s law:

1*sin(theta)=n*sin(theta1)…(2)

for the refraction at second surface:

incident angle=theta1 (as both the panes are parallel to each other)

let refraction angle =theta2

using snell’s law:

n1*sin(theta1)=1*sin(theta2)

using equation 1,

theta2=theta1

hence the light will emerge at the same angle irrespective of value of n

hence angular spread=0 degrees

part b:

for the prism:

for blue light:

index of refraction=n=1.525

for first refraction, let refraction angle be theta.

using snell’s law:

1*sin(46)=1.525*sin(theta)

==>theta=28.145 degrees

this is the angle made by the refracted ray with perpendicular to the surface.

then incident angle at the second surface=90-(180-(60+(90-theta)))

=31.855 degrees

let refracted angle be theta1.

using snell’s law:

1.525*sin(31.855)=1*sin(theta1)

==>theta1=53.597 degrees

for red light:

index of refraction=1.51

for first refraction, let refraction angle be theta.

using snell’s law:

1*sin(46)=1.51*sin(theta)

==>theta=28.449 degrees

this is the angle made by the refracted ray with perpendicular to the surface.

then incident angle at the second surface=90-(180-(60+(90-theta)))

=31.551 degrees

let refracted angle be theta1.

using snell’s law:

1.51*sin(31.551)=1*sin(theta1)

==>theta1=52.196 degrees

so total spread=53.597-52.196=1.401 degrees

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