Apparent depth with multiple indices of refraction

[KBriggs]

Homework Statement

A penny is located at the bottom of a barrel of water 1m deep. There is a 20cm thick layer of oil on top of the water. To an observer at normal incidence, what is the apparent depth of the penny. n for water is 1.33, for oil it is 1.5

Homework Equations

Snell's law, lots of ray diagrams

The Attempt at a Solution

I have already shown that to an observer looking through just the water, the apparent depth is 0.75m. I am just not sure how to generalize this for the other layer. Can someone point me in the right direction?

 

[willem2]

The method is the same, only you have to use Snell's law twice, at the water oil, and oil/air boundaries.

 

[kerimek]

To calculate the apparent depth in this scenario, we can use the concept of apparent depth and Snell's law. The apparent depth is the depth at which an object appears to be located when viewed through a medium with a different refractive index than the actual depth. In this case, we have two different mediums - water with a refractive index of 1.33 and oil with a refractive index of 1.5.

To calculate the apparent depth of the penny in this scenario, we can use the following formula:

Apparent depth = Actual depth / Refractive index of medium

First, let's calculate the apparent depth through the layer of water. Using the given information, the actual depth of the penny is 1m and the refractive index of water is 1.33. Plugging these values into the formula, we get:

Apparent depth = 1m / 1.33 = 0.75m

Thus, as you have already shown, to an observer looking through just the water, the apparent depth of the penny is 0.75m.

Now, to calculate the apparent depth through the layer of oil, we can use the same formula. The actual depth of the penny is still 1m, but the refractive index of oil is 1.5. Plugging these values into the formula, we get:

Apparent depth = 1m / 1.5 = 0.67m

Therefore, to an observer looking through both the water and the oil, the apparent depth of the penny is 0.67m. This means that the penny will appear closer to the surface of the oil than it actually is.

To better understand this concept, we can also use ray diagrams to visualize the path of light as it travels through the different layers. As light travels from water to oil, it bends away from the normal line due to the higher refractive index of oil. This results in the penny appearing closer to the surface of the oil.

In summary, the apparent depth of the penny in this scenario is 0.75m when viewed through just the water, and 0.67m when viewed through both the water and the oil. This concept can be generalized to scenarios with multiple layers of different refractive indices by using the formula mentioned above. I hope this helps!