A cup with refractive index n2 = 1.8 has outer radius b = 6.0 cm and inner radius a = 4.0 cm. It is filled with a liquid with refractive index n3 = 1.3. In the surrounding medium with n1 = 1.0,a ray of light travels along a trajectory that is at a perpendicular distance y from the center of the cup. The ray enters the cup and, reaching the liquid interface at the critical angle, undergoes total internal reflection. Determine the distance y
To determine the distance y, we need to find the critical angle of the light ray entering the cup.
The critical angle can be calculated using Snell's law:
n1 * sin(θ1) = n3 * sin(θ3)
where n1 is the refractive index of the surrounding medium, θ1 is the angle of incidence, n3 is the refractive index of the liquid, and θ3 is the angle of refraction (which is 90 degrees for total internal reflection).
Given:
n1 = 1.0 (refractive index of surrounding medium)
n3 = 1.3 (refractive index of liquid)
We can rearrange Snell's law to solve for sin(θ1):
sin(θ1) = (n3 / n1) * sin(θ3)
= (1.3 / 1.0) * sin(90)
≈ 1.3
Now, we can use the inverse sine function to find the angle θ1:
θ1 = sin^(-1)(1.3)
≈ 90 degrees
Since the light ray undergoes total internal reflection at the liquid interface, the angle of incidence is equal to the critical angle. Therefore, θ1 ≈ 90 degrees.
Now, we can use simple trigonometry to find the perpendicular distance y:
y = b - a
where b is the outer radius of the cup and a is the inner radius.
Given:
b = 6.0 cm
a = 4.0 cm
y = 6.0 cm - 4.0 cm
= 2.0 cm
Therefore, the distance y is 2.0 cm.
To determine the distance y, we need to consider the conditions for total internal reflection at the liquid interface within the cup. Let's break down the problem step by step:
Step 1: Determine the critical angle at the liquid interface.
The critical angle (θc) is the angle of incidence at which light is refracted at an angle of 90 degrees. It can be found using Snell's Law:
n1 * sin(θ1) = n3 * sin(θc)
Here, n1 is the refractive index of the surrounding medium, and n3 is the refractive index of the liquid.
Given:
n1 = 1.0
n3 = 1.3
We can rearrange the equation to solve for sin(θc):
sin(θc) = (n1 / n3) * sin(θ1)
Since θ1 is at a perpendicular distance y from the center of the cup, it creates a right angle with the perpendicular to the liquid interface. Therefore, sin(θ1) = (y / b), where b is the outer radius of the cup.
Step 2: Calculate the critical angle.
Plugging in the values, we have:
sin(θc) = (n1 / n3) * (y / b)
Substituting the given values, we get:
sin(θc) = (1.0 / 1.3) * (y / 6.0)
Step 3: Calculate the distance y.
We know that sin(θc) = 1 when total internal reflection occurs. Therefore, the equation becomes:
1 = (1.0 / 1.3) * (y / 6.0)
Simplifying the equation:
6 = 1.3 * y / 6.0
Multiplying both sides by 6.0:
36 = 1.3 * y
Dividing both sides by 1.3:
y = 36 / 1.3
Calculating the distance y:
y ≈ 27.69 cm
Therefore, the distance y is approximately 27.69 cm.