You're a glaciologist researching the impact of climate...

You're a glaciologist researching the impact of climate change. On the surface of a glacier you're studying the wind has eroded a groove that forms a half cylinder $86.2 \mathrm{~cm}$ in diameter. Looking into this groove you see an air bubble that appears to be at a depth of $75.0 \mathrm{~cm}$ inside the ice. How far inside the ice is the bubble actually located? Hint: Consult Table $30.1$ for the refractive index of ice.

Key Concepts

Apparent Depth versus Actual Depth

The concept of apparent depth versus actual depth arises from the refraction of light at the boundary between two media. When light rays exit a medium with a higher refractive index (like ice) into a medium with a lower refractive index (like air), they bend away from the normal, causing objects to appear closer to the surface than they truly are. This effect is quantified by the refractive index, and understanding it is essential for accurately determining the position of objects as seen through refractive materials.

Refraction

Refraction is the bending of light as it passes from one medium to another with a different optical density. This bending is due to the change in the speed of light in different materials and is governed by Snell’s Law. Refraction affects how objects are perceived when viewed through a medium, leading to optical illusions such as objects appearing to be at a different depth than they actually are.

Index of Refraction

The index of refraction is a dimensionless number that characterizes how much light slows down in a given medium compared to its speed in a vacuum. It plays a crucial role in determining the angle at which light is refracted using Snell’s Law. The value of the refractive index directly influences the relationship between the apparent depth and the true depth of an object when observed from a different medium.

Transcript

00:01 So in this problem, number 36, another variation of example 314 in the text.

00:08 You're a glaciologist researching the impact climate change, and you see a bubble under the surface of the ice, but you are looking at it through a groove that forms a half cylinder.

00:19 It says 86 .2 centimeters in diameter.

00:22 So that would mean the radius of that half bubble is, or that half cylinder is 43 .1 centimeters.

00:31 And optically, that would be a positive value because the bubble is to the right of the surface in my drawing, and that's a convex surface.

00:44 So the radius would be a positive number.

00:47 The image that you see through the ice appears to be at a depth of 75 centimeters.

00:55 So in this case, this is the image distance.

00:59 And it would be negative because the object is clearly under the ice as well, somewhere else, most likely over here somewhere.

01:10 But when the image and the object are on the same side of the surface, it's not really a lens in this case, but it's sort of acting as a lens.

01:20 The image distance is negative.

01:23 So we'll remember that negative sign there.

01:27 And we know it's a virtual image.

01:29 So we're going to use the equation derived in the text for a interface between curved surfaces of different indices of refraction, which is the first index of refraction divided by the object distance plus the second index of refraction divided by the image distance is equal to the difference in indices divided by the radius of that curved surface.

01:56 So we are starting from ice.

01:58 The index of refraction for ice, looking it up in the table, is 1 .309.

02:04 And then the interface goes to an index of refraction of air, which is 1.

02:11 So because we start here, we'll call this n1, and we go to air.

02:17 This is n2...

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