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?
n1 sinθ1 = n2 sinθ2
do for each, n for air is 1
To solve this problem, we need to apply Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the materials involved.
According to Snell's law, the relationship between the angles of incidence and refraction and the indices of refraction can be expressed as follows:
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.
Let's analyze the given information step by step:
1. In the first scenario, the angle of incidence on the sheet 1 - sheet 2 interface is 19.8°, and the angle of refraction in sheet 2 is 31.7°.
Using Snell's law:
n1 * sin(19.8°) = n2 * sin(31.7°)
2. In the second scenario, the angle of incidence on the sheet 3 - sheet 2 interface is the same as before (19.8°), and the angle of refraction in sheet 3 is 36.7°.
Using Snell's law again, but substituting the previously known values:
n1 * sin(19.8°) = n2 * sin(31.7°)
n2 * sin(36.7°) = n3 * sin(theta3)
3. Now, we want to find the expected angle of refraction (theta3) when sheet 1 is on top of sheet 3, with the same angle of incidence (19.8°).
We can rearrange the second equation:
sin(theta3) = (n2 * sin(36.7°)) / n3
4. To solve for theta3, we need to find the ratio n2 / n3. We already have an equation involving n1 and n2 from the first scenario:
n1 * sin(19.8°) = n2 * sin(31.7°)
Rearranging this equation:
n2 / n1 = sin(19.8°) / sin(31.7°)
5. Now, we substitute the expression for n2 / n1 into the equation from step 3:
sin(theta3) = [(sin(19.8°) / sin(31.7°)) * sin(36.7°)] / n3
Finally, we can solve for theta3 by taking the inverse sine of both sides of the equation:
theta3 = arcsin([(sin(19.8°) / sin(31.7°)) * sin(36.7°)] / n3)
Therefore, following these steps, we can find the expected angle of refraction in sheet 3 when sheet 1 is on top of sheet 3.