The light beam shown in the figure on the right
makes an angle of 20.0° with the normal line NN’ in
the linseed oil. Determine the angles θ and θ’. (Note:
The index of refraction of linseed oil is 1.48.)
To determine the angles θ and θ' in the given situation, we can use Snell's law which relates the angles of incidence and refraction to the indices of refraction of the two materials involved.
Snell's Law states:
n1 * sin(θ1) = n2 * sin(θ2)
Where:
n1 is the index of refraction of the initial material (incident medium)
θ1 is the angle of incidence
n2 is the index of refraction of the second material (refracted medium)
θ2 is the angle of refraction
In this case, the incident medium is air (assumed to have an index of refraction of 1), and the refracted medium is linseed oil with an index of refraction of 1.48.
Given:
Angle of incidence, θ1 = 20.0°
Index of refraction of linseed oil, n2 = 1.48
Using Snell's law, we can solve for θ2:
1 * sin(20.0°) = 1.48 * sin(θ2)
sin(θ2) = (1 * sin(20.0°)) / 1.48
sin(θ2) ≈ 0.227
To find θ2, we need to take the inverse sine (sin^-1) of both sides:
θ2 = sin^-1(0.227)
θ2 ≈ 13.0°
So, the angle of refraction θ2 is approximately 13.0°.
Now, let's find θ':
In the figure, θ' is the angle between the refracted ray and the normal line in linseed oil.
Since the angle of incidence θ1 is given, we can use the relationship between θ' and θ1:
θ' = 90° - θ1
θ' = 90° - 20.0°
θ' = 70.0°
Therefore, the angle θ' is equal to 70.0°.
To determine the angles θ and θ', 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(θ₁) = n₂ * sin(θ₂),
where:
- n₁ and n₂ are the indices of refraction of the first and second media, respectively,
- θ₁ is the angle of incidence of the light ray with respect to the normal in the first medium,
- θ₂ is the angle of refraction of the light ray with respect to the normal in the second medium.
In this case, the first medium is the air (or vacuum) with an index of refraction of approximately 1, and the second medium is the linseed oil with an index of refraction of 1.48.
Given that the angle of incidence (θ₁) is 20.0°, we can use Snell's Law to find θ₂:
1 * sin(20.0°) = 1.48 * sin(θ₂).
Let's solve for θ₂:
θ₂ = arcsin((1 * sin(20.0°)) / 1.48).
Using a calculator, we find that θ₂ is approximately 13.20°.
Now, to find θ', which is the angle of incidence of the refracted ray with respect to the normal in the linseed oil, we can use the fact that the angles of incidence and refraction are complementary:
θ' = 90° - θ₂.
Therefore, θ' = 90° - 13.20° = 76.80°.
Thus, the angles θ and θ' are approximately 13.20° and 76.80°, respectively.
sinθ/sinθ' = 1.48
sinθ' = sin20°/1.48 = 0.231
θ = 13.36°
Best I can do with no diagram.