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)
This is a Snell's law problem, of course.
For the red light:
1.637 sin Ar = 1.000 sin 40.4
sin Ar = 0.39592
Ar = 23.32 degrees
For the blue light:
1.671 sin Ab = 1.000 sin 40.4
Solve for angle Ab (the blue ray refraction angle) and then compute the dispersion angle Ab - Ar
To calculate the dispersion angle, we need to determine the angles of incidence for red and violet light within the glass block. We can use Snell's law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction of the two media. The equation for Snell's law is:
n₁ * sin(θ₁) = n₂ * sin(θ₂)
where n₁ and n₂ are the indices of refraction of the two media, θ₁ is the angle of incidence, and θ₂ is the angle of refraction.
First, let's calculate the angle of incidence for red light within the glass block. The index of refraction for red light in glass is 1.637. We can rearrange Snell's law to solve for the angle of incidence:
sin(θ₁) = (n₂ / n₁) * sin(θ₂)
Since the light is incident at 40.40°, we can use trigonometric identity to determine that sin(40.40°) = 0.6443. Substituting the values into the equation, we get:
sin(θ₁) = (1.671 / 1.637) * sin(40.40°)
sin(θ₁) = 1.0208 * 0.6443
sin(θ₁) = 0.6579
To determine the angle of incidence, we can take the inverse sine of 0.6579:
θ₁ = sin⁻¹(0.6579)
θ₁ = 41.69°
Similarly, we can calculate the angle of incidence for violet light using the index of refraction for violet light in glass, which is 1.671. Substituting the values into Snell's law, we have:
sin(θ₁) = (1.671 / 1.637) * sin(θ₂)
Now, we need to find θ₂, the angle of refraction for the violet light within the glass block. Since we know the angle of incidence of 40.40°, we can use Snell's law again:
sin(θ₁) = (n₂ / n₁) * sin(θ₂)
sin(41.69°) = (1.637 / 1.671) * sin(θ₂)
Solving for sin(θ₂), we get:
sin(θ₂) = (1.637 / 1.671) * sin(41.69°)
sin(θ₂) = 0.9804 * sin(41.69°)
sin(θ₂) = 0.6452
Taking the inverse sine of 0.6452, we find:
θ₂ = sin⁻¹(0.6452)
θ₂ = 40.59°
Finally, we can calculate the dispersion angle by taking the difference between the angles of incidence for red and violet light:
Dispersion angle = θ₁ - θ₂
Dispersion angle = 41.69° - 40.59°
Dispersion angle ≈ 1.10°
Therefore, the dispersion angle (i.e., the difference between the directions of red and violet light within the glass block) is approximately 1.10°.