Can anyone help me here? I have no idea where to start!
Light in a vacuum is incident on a transparent glass slab. The angle of incidence is 35.1°. The slab is then immersed in a pool of liquid. When the angle of incidence for the light striking the slab is 21.1°, the angle of refraction for the light entering the slab is the same as when the slab was in a vacuum. What is the index of refraction of the liquid?
To find the index of refraction of the liquid, we can use the Snell's law equation, which states:
n1 * sin(theta1) = n2 * sin(theta2)
where:
- n1 is the index of refraction of the medium the light is coming from (in this case, vacuum),
- n2 is the index of refraction of the medium the light is entering (in this case, the glass slab and then the liquid),
- theta1 is the angle of incidence,
- theta2 is the angle of refraction.
Given:
- Angle of incidence (theta1) when light is incident on the transparent glass slab is 35.1°.
- Angle of incidence (theta1) when light is incident on the glass slab immersed in liquid is 21.1°.
- Angle of refraction (theta2) remains the same for both situations.
Now, let's solve this step by step:
Step 1: Find the index of refraction of the glass slab.
To find the index of refraction of the glass slab, we need the angle of refraction (theta2) when the light enters the slab from vacuum. According to the given data, the angle of refraction remains the same in both the vacuum and the liquid. So, the index of refraction of the glass slab is the same as when the slab was in a vacuum. Let's denote this as n1.
Step 2: Write down the Snell's law equation for the first situation (light incident on the glass slab from vacuum):
n1 * sin(theta1) = n2 * sin(theta2)
Step 3: Substitute the given values into the equation:
- theta1 = 35.1°
- theta2 (angle of refraction in vacuum) = 35.1° (given that the angle of incidence is the same as the angle of refraction in vacuum)
The equation becomes:
n1 * sin(35.1°) = n2 * sin(35.1°)
Step 4: Simplify the equation:
n1 = n2
Step 5: Find the index of refraction of the liquid.
Now, let's consider the second situation where the angle of incidence is 21.1°. Substituting the values into the Snell's law equation, we get:
n1 * sin(theta1) = n2 * sin(theta2)
Since the angle of refraction in the liquid is the same as the angle of refraction in vacuum (35.1°), we can rewrite the equation as:
n1 * sin(21.1°) = n2 * sin(35.1°)
Step 6: Solve for n2:
n2 = (n1 * sin(21.1°)) / sin(35.1°)
Step 7: Substitute the value of n1:
n2 = (n2 * sin(21.1°)) / sin(35.1°)
Step 8: Solve for n2:
n2 * sin(35.1°) = n2 * sin(21.1°)
Step 9: Cancel out n2 on both sides:
sin(35.1°) = sin(21.1°)
Step 10: Solve for sin(35.1°) and sin(21.1°) using a calculator or math software.
Once you have the values for sin(35.1°) and sin(21.1°), you can solve for n2 using the equation:
n2 = (sin(21.1°) * n1) / sin(35.1°)
By substituting the value of n1 and the calculated values of sin(21.1°) and sin(35.1°), you can find the index of refraction (n2) of the liquid.
Sure, I can help you with that! To find the index of refraction of the liquid, we can use Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the two media involved.
Snell's law is written as:
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.
In this case, we can start by considering the situation when the glass slab is in a vacuum. Given that the angle of incidence is 35.1° and the angle of refraction is the same, we can write the equation as:
n_glass * sin(35.1°) = 1 * sin(35.1°)
Since the index of refraction for vacuum or air is approximately equal to 1, we can simplify the equation to:
n_glass * sin(35.1°) = sin(35.1°)
Now, let's consider the situation when the glass slab is immersed in the liquid. Given that the angle of incidence is 21.1° and the angle of refraction is the same as before, we can write the equation as:
n_glass * sin(21.1°) = n_liquid * sin(21.1°)
Now, we have two equations:
1. n_glass * sin(35.1°) = sin(35.1°) [from the vacuum case]
2. n_glass * sin(21.1°) = n_liquid * sin(21.1°) [from the liquid case]
To find the index of refraction of the liquid (n_liquid), we can divide equation 2 by equation 1:
(n_glass * sin(21.1°)) / (n_glass * sin(35.1°)) = (n_liquid * sin(21.1°)) / sin(35.1°)
sin(21.1°) cancels out, and we are left with:
n_liquid = sin(21.1°) / sin(35.1°)
Using a scientific calculator, we can calculate this value.
So, to summarize:
1. Calculate sin(21.1°) and sin(35.1°).
2. Divide sin(21.1°) by sin(35.1°).
This will give you the value of the index of refraction of the liquid.