What is the index of refraction of a material if the angle of incidence in air is 50 degrees and the angle of refraction in the material is 40 degrees? Please show steps.
Why did the light bend? Because it wanted to make a good impression on the material! Okay, let's bring out our clown calculator and calculate the refractive index.
The index of refraction (n) can be found using Snell's law, which states:
n₁sin(θ₁) = n₂sin(θ₂)
Here, n₁ is the index of refraction of air, θ₁ is the angle of incidence in air, n₂ is the index of refraction of the material, and θ₂ is the angle of refraction in the material.
Given:
θ₁ = 50 degrees
θ₂ = 40 degrees
We know that the index of refraction of air is approximately 1, as air isn't a fan of bending light. So, let's rearrange the formula and solve for n₂:
n₁sin(θ₁) = n₂sin(θ₂)
1 * sin(50) = n₂ * sin(40)
sin(50) = n₂ * sin(40)
n₂ = sin(50) / sin(40)
Now, let's put our calculations into action:
n₂ = sin(50) / sin(40)
n₂ ≈ 0.766 / 0.642
Hold your balloons; the clown calculator is working its magic!
n₂ ≈ 1.194
So, the index of refraction of the material is approximately 1.194. Remember, you can always double-check my calculations, but I guarantee you won't find any funny business here.
To calculate the index of refraction, 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 velocities:
n1 * sin(angle of incidence) = n2 * sin(angle of refraction)
Where:
n1 = index of refraction of the medium of incidence (in this case, air)
n2 = index of refraction of the medium of refraction (the material)
Given:
Angle of incidence in air = 50 degrees
Angle of refraction in the material = 40 degrees
Step 1: Convert the angles from degrees to radians.
angle of incidence in radians = 50 degrees * (π / 180) = 0.8727 radians
angle of refraction in radians = 40 degrees * (π / 180) = 0.6981 radians
Step 2: Apply Snell's law to find the index of refraction.
n1 * sin(0.8727) = n2 * sin(0.6981)
Step 3: Solve for n2.
n2 = (n1 * sin(0.8727)) / sin(0.6981)
Step 4: Calculate the value of n2.
n2 = (1 * sin(0.8727)) / sin(0.6981)
n2 ≈ 1.4607
Therefore, the index of refraction of the material is approximately 1.4607.
To find the index of refraction of a material, you can use Snell's law, which relates the angles of incidence and refraction to the index of refraction of the two materials involved.
Snell's law is expressed as:
n₁sinθ₁ = n₂sinθ₂
Where:
- n₁ is the index of refraction of the first medium (in this case air)
- θ₁ is the angle of incidence in the first medium
- n₂ is the index of refraction of the second medium (the material)
- θ₂ is the angle of refraction in the second medium
In this problem, we have:
- Angle of incidence in air (θ₁) = 50 degrees
- Angle of refraction in the material (θ₂) = 40 degrees
We need to solve for the index of refraction of the material (n₂).
Here are the steps to find n₂:
Step 1: Convert the angles to radians
- Convert the angle of incidence from degrees to radians: θ₁_rad = θ₁ * (π/180)
- Convert the angle of refraction from degrees to radians: θ₂_rad = θ₂ * (π/180)
Step 2: Apply Snell's law
- Divide sinθ₁ by sinθ₂ to isolate the index of refraction of the material (n₂)
- n₂ = n₁ * (sinθ₁ / sinθ₂)
Step 3: Substitute the values
- Substitute the given values of θ₁_rad and θ₂_rad into the equation:
- n₂ = n₁ * (sin(θ₁_rad) / sin(θ₂_rad))
Step 4: Calculate the index of refraction
- Plug in the values and evaluate the equation to find n₂:
- n₂ = n₁ * (sin(50°*(π/180)) / sin(40°*(π/180)))
Step 5: Solve for n₂
- Use a scientific calculator to calculate the sin values and then substitute into the equation to find n₂.
Once you've completed these steps, you should have the value of the index of refraction of the material.
ni sin i = nr sin r
1 sin 50 = nr sin 40
nr = sin 50 / sin 40