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).
How wide is the beam when it reaches the bottom of the block, as measured along the bottom surface of the block?
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 dispersion angle, we can use Snell's Law, which relates the angles of incidence and refraction as well as the indices of refraction. The equation is:
n₁sinθ₁ = n₂sinθ₂
Where:
n₁ is the index of refraction of the glass for red light (1.637)
n₂ is the index of refraction of the glass for violet light (1.671)
θ₁ is the incident angle (40.40°)
First, we need to find the refracted angles for red and violet light. Rearrange Snell's Law to solve for θ₂:
θ₂ = sin^(-1)((n₁/n₂) * sinθ₁)
For red light:
θ₁ = 40.40°
n₁ = 1.637
n₂ = 1.671
θ₂(red) = sin^(-1)((1.637/1.671) * sin(40.40°))
Calculate the dispersion angle by subtracting these two angles:
Dispersion angle = θ₂(violet) - θ₂(red)
Next, let's calculate the width of the beam when it reaches the bottom of the block. This can be done using trigonometry and the thickness of the block.
Given:
Thickness of the glass block = 11.8 cm
Incident angle = 40.40°
We can use the formula:
Width of the beam = Thickness of the block * tan(incident angle)
Width of the beam = 11.8 cm * tan(40.40°)
Finally, to calculate the distance between the exiting red and violet rays (the thickness of the rainbow), we can use the same formula for calculating the width of the beam. However, this time we need to use the dispersion angle instead of the incident angle:
Thickness of the rainbow = Thickness of the block * tan(dispersion angle)
Now you have the steps to calculate the dispersion angle, the width of the beam when it reaches the bottom of the block, and the distance between the exiting red and violet rays. Plug in the given values into the equations to find the actual numerical values.