Problem 47 - University Physics with Modern Physics

Question

A thin layer of ice (\(n\) = 1.309) floats on the surface of water (\(n\) = 1.333) in a bucket. A ray of light from the bottom of the bucket travels upward through the water. (a) What is the largest angle with respect to the normal that the ray can make at the ice-water interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

Step-by-Step Solution

Answer

(a) Approximately 50.4°; (b) Approximately 48.8° after the ice melts.

Step 1: Understand the Concept

The problem asks for the largest angle at which a ray of light can pass from one medium to another without being totally internally reflected. This is known as the critical angle, which occurs when light is at the boundary between refraction and reflection.

Step 2: Determine the Interface

Identify the interfaces involved: the light travels from water (\(n = 1.333\)) to ice (\(n = 1.309\)), and then from ice to air (\(n = 1.00\)). Calculate the critical angle for both interfaces.

Step 3: Use Snell's Law

Snell's Law is given by \(n_1 \sin(\theta_1) = n_2 \sin(\theta_2)\). To find the critical angle, we set \(\theta_2 = 90^\circ\) because beyond this angle, total internal reflection occurs. Hence, the equation becomes \(\sin(\theta_c) = \frac{n_2}{n_1}\).

Step 4: Calculate Critical Angle at Ice-Water Interface Before Melting

For the ice-water interface, the ray of light is going from water to ice. The critical angle (\(\theta_{c1}\)) can be calculated using \(\sin(\theta_{c1}) = \frac{n_{ice}}{n_{water}} = \frac{1.309}{1.333}\). Calculate \(\theta_{c1}\) using this ratio.

Step 5: Calculate Critical Angle at Ice-Air Interface

For the ice-air interface, the ray goes from ice to air. The critical angle (\(\theta_{c2}\)) is found using \(\sin(\theta_{c2}) = \frac{n_{air}}{n_{ice}} = \frac{1.00}{1.309}\). Calculate \(\theta_{c2}\) using this ratio.

Step 6: Determine the Largest Angle Before Ice Melts

The angle that limits light exiting from water through ice into air is determined by the smaller critical angle from the steps above. Compare the angles \(\theta_{c1}\) and \(\theta_{c2}\) and choose the smaller as the largest angle of incidence for light passing out into the air.

Step 7: Angle After Ice Melts

Once the ice melts, light directly travels from water to air. Calculate the critical angle using \(\sin(\theta_{c}) = \frac{n_{air}}{n_{water}} = \frac{1.00}{1.333}\). This gives the new maximum angle.

Key Concepts

Snell's Law

Snell's Law is fundamental in understanding the refraction of light when it moves between different media. It explains why light bends when crossing from one substance to another. The law is expressed by the equation \(n_1 \sin(\theta_1) = n_2 \sin(\theta_2)\). Here, \(n_1\) and \(n_2\) are the refractive indices of the two media, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively.

One practical application of Snell's Law is in calculating the critical angle for total internal reflection, a crucial concept in optics. For a ray traveling from a denser medium to a less dense one, if it hits the boundary at an angle greater than this critical angle, it will not pass through but instead reflect back into the denser medium.

• The denser medium has a higher refractive index (\(n\)).
• Total internal reflection occurs when \(\theta_2\) is 90 degrees, making the refracted ray skim along the boundary.

Total Internal Reflection

Total Internal Reflection is an intriguing phenomenon where light is completely reflected back into a medium rather than passing through it. This effect is observed when a ray of light moves from a medium with a higher refractive index to one with a lower refractive index.

The critical angle is key to understanding total internal reflection. It is the maximum angle of incidence in the denser medium beyond which the light cannot refract into the less dense medium. Instead, it is entirely reflected back inside. This happening is not just a peculiarity; it is widely used in designing instruments like fiber optics, binoculars, and even periscopes.

• The critical angle depends on the refractive indices of the two media in contact.
• If the incident angle is greater than the critical angle, the light doesn't escape.

Refraction of Light

Refraction is the bending of light as it passes from one medium to another with a different refractive index. This occurs because the speed of light changes in different materials, altering its path.

When light enters a denser medium (e.g., from air to water), it slows down and bends towards the normal – an imaginary line perpendicular to the surface at the point of incidence. Conversely, transitioning into a less dense medium causes the ray to speed up and bend away from the normal. Snell's Law helps calculate this bending precisely using the media's refractive indices.

• Bending is more pronounced with larger differences in refractive indices.
• Light waves are generally fastest in vacuum and slowest in solid materials.