A narrow ray of yellow light from glowing sodium (λ0 = 589 nm) traveling in air strikes a smooth surface of water at an angle of θi = 40.5°. Determine the angle of refraction θr.
To determine the angle of refraction (θr), we can use Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the two media.
Snell's law states: n1 * sin(θi) = n2 * sin(θr)
Where:
- n1 is the refractive index of the incident medium (in this case, air)
- θi is the angle of incidence
- n2 is the refractive index of the refractive medium (in this case, water)
- θr is the angle of refraction
We know that the wavelength of the yellow light from glowing sodium (λ0) is 589 nm. To find the refractive index for each medium, we can use the equation:
Refractive index (n) = c/v
Where:
- c is the speed of light in vacuum (3 x 10^8 m/s)
- v is the speed of light in the medium
For air, the speed of light is nearly the same as in vacuum, so we can consider n1 ≈ 1.
For water, the speed of light is approximately 2.25 x 10^8 m/s.
Let's calculate the angle of refraction using the given information:
Step 1: Calculate the refractive index of water (n2):
n2 = c/v = (3 x 10^8 m/s) / (2.25 x 10^8 m/s) = 1.33
Step 2: Apply Snell's law to calculate the angle of refraction (θr):
n1 * sin(θi) = n2 * sin(θr)
1 * sin(40.5°) = 1.33 * sin(θr)
Now, we can solve for θr.
θr = arcsin((1 * sin(40.5°)) / 1.33)
Using a calculator, we find θr ≈ 30.6°.
Therefore, the angle of refraction (θr) is approximately 30.6°.