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 can use Snell's Law and the concept of critical angle.

Snell's Law states that the ratio of the sine of the incident angle to the sine of the refracted angle is equal to the ratio of the refractive indices of the two media:

n1 * sin(θ1) = n2 * sin(θ2)

In this case, the incident medium is the surrounding medium (n1 = 1.0) and the refracted medium is the cup material (n2 = 1.8). The incident angle θ1 is given such that the trajectory is at a perpendicular distance y from the center of the cup. The refracted angle θ2 can be determined using the critical angle.

The critical angle for total internal reflection is given by:

sin(θc) = n2 / n3

where n3 is the refractive index of the liquid filled in the cup.

Since the ray undergoes total internal reflection at the liquid interface, the angle of incidence at the interface (θ1) is equal to the critical angle (θc):

θ1 = θc

Now, we can substitute these values into Snell's Law to find the angle of refraction (θ2):

n1 * sin(θ1) = n2 * sin(θ2)
1.0 * sin(θ1) = 1.8 * sin(θc)
sin(θ1) = (1.8 / 1.0) * sin(θc)
sin(θ1) = 1.8 * (n2 / n3)

The equation above relates the angle of incidence at the cup interface (θ1) with the critical angle (θc) and the refractive indices of the cup material (n2) and liquid (n3).

To find the distance y, we need to solve for sin(θ1) and then use its inverse to obtain the angle θ1. The distance y can then be calculated using trigonometry.

Note: Alternatively, you can also use the concepts of refraction and total internal reflection to calculate the distance y using the angles of incidence and refraction.