Question

(a) A tank containing methanol has walls 2.50 cm thick made of glass of refractive index 1.550. Light from the outside air strikes the glass at a 41.3\(^\circ\) angle with the normal to the glass. Find the angle the light makes with the normal in the methanol. (b) The tank is emptied and refilled with an unknown liquid. If light incident at the same angle as in part (a) enters the liquid in the tank at an angle of 20.2\(^\circ\) from the normal, what is the refractive index of the unknown liquid?

Short answer

(a) The angle in methanol is approximately 29.8°. (b) The refractive index of the unknown liquid is about 1.909.

Step-by-step solution

Understand the refraction through glass

For part (a), we have the refractive index of air as \( n_{\text{air}} = 1 \), refractive index of glass \( n_{\text{glass}} = 1.550 \), and the angle of incidence in air \( \theta_{1} = 41.3^\circ \). We need to find the angle of refraction \( \theta_{2} \) in the glass using Snell's law: \[ n_{\text{air}} \sin\theta_{1} = n_{\text{glass}} \sin\theta_{2} \]. Substitution gives:\[ \sin\theta_{2} = \frac{1}{1.550} \sin(41.3^\circ) \].

Solve for the angle in the glass

Solving for \( \theta_{2} \):\[ \sin\theta_{2} = \frac{1}{1.550} (0.6585) \approx 0.4249 \]\( \theta_{2} = \arcsin(0.4249) \approx 25.2^\circ \). This is the angle of refraction in the glass.

Refraction from glass to methanol

Given the refractive index of methanol \( n_{\text{methanol}} = 1.329 \). Use Snell's law again from glass to methanol:\[n_{\text{glass}}\sin\theta_{2} = n_{\text{methanol}} \sin\theta_{3} \]where \( \theta_{2} = 25.2^\circ \) is the angle in the glass and we solve for \( \theta_{3} \) in methanol:\[ \sin\theta_{3} = \frac{1.550 \times 0.4249}{1.329} \approx 0.4962 \]\( \theta_{3} = \arcsin(0.4962) \approx 29.8^\circ \).

Use given conditions for unknown liquid

For part (b), we have \( \theta_{1} = 41.3^\circ \) (incident angle) and \( \theta_{4} = 20.2^\circ \) (refracted angle into unknown liquid). Use Snell's law:\[ n_{\text{air}}\sin\theta_{1} = n_{\text{unknown}} \sin\theta_{4} \]Subsitute known values and solve for \( n_{\text{unknown}} \):\[ 1\sin(41.3^\circ) = n_{\text{unknown}}\sin(20.2^\circ) \]\[ n_{\text{unknown}} = \frac{0.6585}{0.3450} \approx 1.909 \].

Conclusion

For part (a), the angle the light makes with the normal in the methanol is \( 29.8^\circ \). For part (b), the refractive index of the unknown liquid is approximately \( 1.909 \).

Key concepts

Refractive Index

Refractive index is a crucial concept in optics. It indicates how much light bends, or refracts, when entering a different medium. The refractive index, denoted by the symbol "n," is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. This is expressed by the equation \[ n = \frac{c}{v} \]where "c" is the speed of light in a vacuum and "v" is the speed of light in the medium. A higher refractive index means light travels more slowly through the medium, resulting in a greater bending of the light. This property is essential when calculating how light behaves when entering or exiting materials like glass, water, or methanol.
Understanding the refractive index helps us use Snell's Law to predict the path of light in various mediums, making it invaluable in fields that involve optics, such as physics, biology, and even photography.

Angle of Incidence

The angle of incidence is the angle between the incoming light ray and the normal to the surface at the point of contact. In this context, "normal" means perpendicular to the surface. When a light ray strikes a boundary, the angle of incidence helps determine how much the light will reflect or refract. It is usually represented by the Greek letter \( \theta \) with a subscript indicating its position, such as \( \theta_1 \) for the first medium.

• When the angle of incidence is increased, the refraction angle is also likely to change, according to Snell’s Law.
• An incident angle of 0° means the light hits the surface head-on, resulting in no bending.

In everyday examples like sunlight hitting a mirror or glass, understanding this angle helps us predict where the reflected or refracted light will go.

Angle of Refraction

The angle of refraction is the angle formed between the refracted ray and the normal to the surface at the point where the light passes into the new medium. It is determined using Snell's Law, which relates the sine of the angles of incidence and refraction to the refractive indices of the two media:\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]where \( n_1 \) and \( n_2 \) are the refractive indices of the initial and second medium, respectively, while \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction.
This equation is a core tool for predicting how light will behave as it travels between different materials. Knowing the angle of refraction is important in applications ranging from designing lenses to studying the behavior of light in different substances.

Optics

Optics is the branch of physics that studies light and its interactions with different objects. It encompasses a variety of phenomena, including reflection, refraction, and diffraction. These principles explain how light behaves under different circumstances and allow us to design practical applications such as cameras, glasses, and even lasers. In optics, understanding how light bends when passing through various materials is key. This bending, controlled by Snell's Law, is what allows lenses to focus light and for telescopes to magnify distant objects.
The broader field of optics ranges from simple lenses to complex structures like fiber optic cables used in telecommunications, making it essential for various technological advances.

Refraction in Mediums

Refraction is the bending of light as it passes from one medium into another due to a change in its speed. This principle is particularly important when different mediums have different refractive indices. For example, when light moves from air (with a refractive index of 1.0) into water (index of roughly 1.33), its speed decreases and the light ray bends toward the normal.

• Refraction occurs because light changes speed as it travels between mediums with different densities.
• This change in speed causes the light to bend, a phenomenon predicted accurately by Snell's Law.
• Common examples of refraction include the bending of a straw when observed through a glass of water or the mirage effect on hot pavement.

Refraction is a critical concept in optics, helping us to design lenses and optical instruments that correct vision, concentrate light, and study tiny details in research.