A very narrow beam of white light is incident at 40.40° onto the top surface of a rectangular block of flint glass 11.8 cm thick. The indices of refraction of the glass for red and violet light are 1.637 and 1.671, respectively.
Calculate the dispersion angle (i.e., the difference between the directions of red and violet light within the glass block)
okay I can get the dispersion angle using snell law...23.32 and 22.82.
then i am asked to get the width as the rays hit the bottom which I can get using geometry...tan 23.32 = x/ 11.8 and same for the other. answer is .122cm
Then I am asked the following:
When the rainbow-colored beam emerges from the bottom of the block, the exiting rays of light are once again parallel. Calculate the distance between the exiting red and violet rays (i.e., the thickness of the rainbow).
To calculate the distance between the exiting red and violet rays, we can use the concept of dispersion in the glass block.
The dispersion angle is the difference between the directions of red and violet light within the glass block. As you mentioned, the dispersion angles for red and violet light are 23.32° and 22.82°, respectively.
Now, let's consider the geometry of the situation. The incident white light enters the glass block at an angle of 40.40°. After passing through the block, the red and violet rays emerge parallel to each other.
Since the red and violet rays emerge parallel, their angles of incidence on the bottom surface of the glass block must be equal. Let's call this angle of incidence θ.
Using the concept of Snell's Law, we can relate the indices of refraction and the angles of incidence and refraction for each color:
n_red * sin(θ) = n_glass * sin(23.32°) (1)
n_violet * sin(θ) = n_glass * sin(22.82°) (2)
Where n_red, n_violet, and n_glass are the refractive indices of red light, violet light, and glass, respectively.
By rearranging equation (1) and (2) and solving for sin(θ), we can find sin(θ) for both red and violet light.
sin(θ) = (n_red / n_glass) * sin(23.32°) (3)
sin(θ) = (n_violet / n_glass) * sin(22.82°) (4)
Now, we can use the inverse sine function to find θ for red and violet light:
θ_red = arcsin[(n_red / n_glass) * sin(23.32°)] (5)
θ_violet = arcsin[(n_violet / n_glass) * sin(22.82°)] (6)
Finally, the distance between the exiting red and violet rays (thickness of the rainbow) can be calculated using the formula:
Thickness of rainbow = Width of the glass block * tan(θ_red - θ_violet)
In this case, the width of the glass block is 11.8 cm, as you mentioned earlier. Substitute the calculated values of θ_red and θ_violet into the formula to find the thickness of the rainbow.