Three sheets of plastic have unknown indices of refraction. Sheet 1 is placed on top of sheet 2, and a laser beam is directed onto the sheets from above. The laser beam enters sheet 1 and then strikes the interface between sheet 1 and sheet 2 at an angle of 19.8° with the normal. The refracted beam in sheet 2 makes an angle of 31.7° with the normal. The experiment is repeated with sheet 3 on top of sheet 2, and, with the same angle of incidence on the sheet 3–sheet 2 interface, the refracted beam makes an angle of 36.7° with the normal. If the experiment is repeated again with sheet 1 on top of sheet 3, with that same angle of incidence on the sheet 1–sheet 3 interface, what is the expected angle of refraction in sheet 3?

To solve this problem, we can use Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the materials involved. Snell's law is given by:

n1 * sin(theta1) = n2 * sin(theta2)

where n1 and n2 are the indices of refraction of the two materials and theta1 and theta2 are the angles of incidence and refraction, respectively, measured with respect to the normal.

Let's consider the first case, where sheet 1 is on top of sheet 2. We are given that the angle of incidence (theta1) is 19.8° and the angle of refraction (theta2) is 31.7°. We need to find the index of refraction of sheet 2 (n2).

Using Snell's law, we have:

n1 * sin(theta1) = n2 * sin(theta2)

Since we don't know the indices of refraction for any of the sheets, let's assume n1 = 1 (air) for convenience:

1 * sin(19.8°) = n2 * sin(31.7°)

Simplifying, we get:

sin(19.8°) = n2 * sin(31.7°)

To find n2, we divide both sides of the equation by sin(31.7°):

n2 = (sin(19.8°)) / (sin(31.7°))

Using a calculator, we find that n2 ≈ 0.376.

Now, let's move on to the second case, where sheet 3 is on top of sheet 2. We are given that the angle of incidence (theta1) is the same as before, 19.8°, and the angle of refraction (theta2) is 36.7°. We need to find the index of refraction of sheet 3 (n3).

Using Snell's law again, we have:

n2 * sin(theta1) = n3 * sin(theta2)

Since we already know n2 = 0.376, we can substitute this value:

0.376 * sin(19.8°) = n3 * sin(36.7°)

Simplifying, we get:

sin(19.8°) = (n3 * sin(36.7°)) / 0.376

To find n3, we divide both sides of the equation by sin(36.7°) and multiply by 0.376:

n3 = (sin(19.8°) * 0.376) / sin(36.7°)

Using a calculator, we find that n3 ≈ 0.513.

Finally, for the third case, where sheet 1 is on top of sheet 3, we use the same angle of incidence (19.8°) as before and the newly found index of refraction of sheet 3 (n3 ≈ 0.513) to find the expected angle of refraction in sheet 3.

n1 * sin(theta1) = n3 * sin(theta3)

Substituting the values, we get:

1 * sin(19.8°) = 0.513 * sin(theta3)

Simplifying, we have:

sin(19.8°) = 0.513 * sin(theta3)

To find theta3, we take the inverse sine of both sides of the equation:

theta3 = sin^(-1)((sin(19.8°)) / 0.513)

Using a calculator, we find that theta3 ≈ 37.3°.

Therefore, the expected angle of refraction in sheet 3 when sheet 1 is on top is approximately 37.3°.

To solve this problem, 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 indices of refraction of the two media.

Let's denote the indices of refraction of sheet 1, 2, and 3 as n1, n2, and n3 respectively. We are given the following information:

- Angle of incidence on the sheet 1 - sheet 2 interface (θ1) = 19.8°
- Angle of refraction in sheet 2 (θ2) = 31.7°
- Angle of incidence on the sheet 3 - sheet 2 interface (θ3) = Same angle as θ1 = 19.8°
- Angle of refraction in sheet 3 (θ4) = ?

Using Snell's law, we can set up the following equations:

For the sheet 1 - sheet 2 interface:
sin(θ1) / sin(θ2) = n2 / n1

For the sheet 3 - sheet 2 interface:
sin(θ3) / sin(θ4) = n4 / n3

Since the incident angle on the sheet 3 - sheet 2 interface (θ3) and the incident angle on the sheet 1 - sheet 2 interface (θ1) are the same, we can use the same equation to solve for n4 / n3.

Let's substitute the known values into the equations:

sin(19.8°) / sin(31.7°) = n2 / n1
sin(19.8°) / sin(θ4) = n4 / n3

Now we can solve for n2 / n1:

n2 / n1 = sin(19.8°) / sin(31.7°)
n2 = n1 * (sin(19.8°) / sin(31.7°))

Finally, we can substitute this value into the equation for the sheet 3 - sheet 2 interface:

sin(19.8°) / sin(θ4) = (n1 * (sin(19.8°) / sin(31.7°))) / n3

Now, we can solve for θ4 by rearranging the equation:

sin(θ4) = sin(19.8°) / [(n1 * (sin(19.8°) / sin(31.7°))) / n3]
θ4 = arcsin(sin(19.8°) / [(n1 * (sin(19.8°) / sin(31.7°))) / n3])

We have the equation to calculate the expected angle of refraction in sheet 3 (θ4) when sheet 1 is on top of sheet 3.