A diamond (n = 2.42) immersed in glycerin (n = 1.473) liquid. What is the critical angle at the boundary between diamond and glycerin?
It is not clear to me where the light is coming from, internal to the diamond trying to exit, or light in the glycerine trying to enter the diamond.
angle is arc sin (n2 / n1) where n1 is the incident ray, and 2 is the refracted ray
arcsin(1.473/2.42)=37.5 degrees (for light in the diamond trying to escape)
To find the critical angle at the boundary between diamond and glycerin, we can use Snell's law, which states:
n1 * sin(theta1) = n2 * sin(theta2)
where:
n1 is the refractive index of the first medium (diamond),
theta1 is the angle of incidence,
n2 is the refractive index of the second medium (glycerin),
and theta2 is the angle of refraction.
In this case, we want to find the critical angle, which occurs when the angle of refraction is 90 degrees. Therefore, we can set sin(theta2) to 1.
Using this information, we can rearrange the equation to solve for the critical angle (theta1):
sin(theta1) = (n2/n1) * sin(theta2)
sin(theta1) = (1.473/2.42) * 1
Now, we can calculate the critical angle:
sin(theta1) = 0.607
To find the angle (theta1), we can take the inverse sine (sin^-1) of 0.607:
theta1 = sin^-1(0.607)
theta1 ≈ 37.9 degrees
Therefore, the critical angle at the boundary between diamond and glycerin is approximately 37.9 degrees.
To find the critical angle at the boundary between diamond and glycerin, you need to use Snell's Law. Snell's Law relates the angles of incidence and refraction of light as it passes through different mediums.
The formula for Snell's Law is:
n1 * sin(theta1) = n2 * sin(theta2)
Where:
- n1 and n2 are the refractive indices of the two mediums (in this case, diamond and glycerin)
- theta1 and theta2 are the angles of incidence and refraction, respectively
In this scenario, the light is passing from diamond (n1) to glycerin (n2).
Given:
n1 (diamond) = 2.42
n2 (glycerin) = 1.473
The critical angle occurs when the refracted angle (theta2) is 90 degrees. At the critical angle, the angle of incidence (theta1) will be the maximum value for total internal reflection to occur.
Rearranging Snell's Law to solve for the angle of incidence:
sin(theta1) = (n2 / n1) * sin(theta2)
Using the values:
sin(theta1) = (1.473 / 2.42) * sin(90 degrees)
Calculating sin(90 degrees) gives a value of 1, so:
sin(theta1) = (1.473 / 2.42) * 1
sin(theta1) ≈ 0.6078
Now, to find the critical angle, you need to take the inverse sine (also known as arcsine) of this value:
theta1 (critical angle) = arcsin(0.6078)
Using a calculator or a mathematical software, the critical angle is approximately 37 degrees.