Figure P 35.45 shows a desk ornament globe containing a...

Figure $P 35.45$ shows a desk ornament globe containing a photograph. The flat photograph is in air, inside a vertical slot located behind a water-filled compartment having the shape of one half of a cylinder. Suppose you are looking at the center of the photograph and then rotate the globe about a vertical axis. You find that the center of the photograph disappears when you rotate the globe beyond a certain maximum angle (Fig. P35.45b). Account for this phenomenon and calculate the maximum angle. Briefly describe what you would see when you turn the globe beyond this angle. (FIGURE A,AND B, CANT COPY)

Key Concepts

Optical Geometry in Curved Surfaces

The behavior of light in systems with curved refracting surfaces, such as a half-cylindrical water compartment, is governed by geometric optics. The curvature affects the angles at which light reaches the boundary and thus modifies the apparent field of view and the conditions necessary for total internal reflection. This concept is essential for calculating the maximum rotation angle beyond which the photograph is no longer visible and for understanding the nature of the partially visible, distorted image seen when the globe is rotated further.

Total Internal Reflection

Total internal reflection occurs when light, traveling in a medium of higher refractive index, hits the boundary with a lower refractive index at an angle greater than the critical angle, resulting in the light being completely reflected back into the medium instead of being refracted. In the context of the problem, as the globe rotates, the light rays from the center of the photograph eventually meet the water-air interface at angles exceeding the critical angle, leading them to be internally reflected and thereby making the photograph seem to vanish.

Snell’s Law

This fundamental law of optics governs how light rays change direction when passing between two media with different refractive indices. It is expressed mathematically by the relation n?sin?? = n?sin??, where n? and n? are the refractive indices of the two media and ?? and ?? are the angles of incidence and refraction, respectively. In the problem at hand, Snell’s law is crucial in determining how light from the photograph bends as it exits the water-filled compartment and enters the surrounding air.

Critical Angle

The critical angle is defined as the angle of incidence in a denser medium above which the refracted ray in the less dense medium would be at 90° relative to the normal at the interface. It can be calculated using the formula sin(?c) = n?/n?, where n? is the refractive index of the rarer medium (air) and n? is that of the denser medium (water). This concept explains why, beyond a certain viewing angle, the light rays from the photograph no longer emerge from the water, causing the image of the photograph’s center to disappear.

Please give Ace some feedback