Problem 7 - Master Resource Book in JEE Main Physics

Question

A fish is a little away below the surface of a lake. If the critical angle is \(49^{\circ}\), then the fish could see things above water surface within an angular range of \(\theta^{\circ}\) where (a) \(\theta=49^{\circ}\) (b) \(\theta=98^{\circ}\) (c) \(\theta=24 \frac{1}{4}\) [d) \(\theta=90^{\circ}\)

Step-by-Step Solution

Answer

The answer is (b) \(\theta=98^{\circ}\).

Step 1: Understand Critical Angle

The critical angle is the angle of incidence above which total internal reflection occurs. It is the largest angle at which light can escape from water to air. Given that the critical angle is \(49^{\circ}\), light can escape to air at this angle or less.

Step 2: Determine Angular Range

A fish below the surface can see light coming from above water within the cone defined by the critical angle around the vertical. This cone forms a circle on the water surface, and its angular range around the horizon totals \(2 \times 49^{\circ} = 98^{\circ}\).

Step 3: Identify Correct Option

Since the angular range of sight for the fish is \(98^{\circ}\), the correct choice out of the given options is (b) \(\theta=98^{\circ}\).

Key Concepts

Total Internal Reflection

Total internal reflection is an intriguing optical phenomenon. It occurs when a wave, like light, hits the boundary of a medium at an angle greater than the medium's critical angle.
Imagine light traveling through water and reaching the surface to enter the air. If the light strikes the water-air boundary at a sharp angle, beyond its critical angle, it reflects back into the water instead of passing into the air. This reflection is what we call total internal reflection.

• Critical angle is key: It is the angle of incidence above which total internal reflection takes place.
• Inside the medium: All the light reflects back without any absorption or bending into the next medium.
• Applications: This principle is used in fiber optics and some binoculars.

This concept allows phenomena like the shimmering path of sunlight on water or the internal mirage in a fish tank.

Angle of Incidence

The angle of incidence is a fundamental concept in optics. It refers to the angle between an incoming ray of light and a line perpendicular to the surface it hits.
This angle determines how the light will behave upon reaching the surface. Will it bounce back, bend, or disappear into the new medium?

• Measurement: Measured from a perpendicular draw at the point of contact or interface.
• Critical role: Helps determine when total internal reflection occurs.
• Influences refraction: Affects how much light bends when passing from one medium to another.

Understanding this angle helps predict and explain different optical effects, including the shimmering water's surface. In the fish scenario, it helps understand what the fish sees from beneath the surface.

Light Refraction

Light refraction is the bending of light as it moves from one medium to another, like from air to water. This bending occurs because light travels at different speeds in different media.
The amount of bending depends on the angle of incidence and the optical properties of the two media.

• Snell's Law: Governs the relationship between the angles of incidence and refraction across different media.
• Critical importance: Essential in lenses that correct vision or magnify objects.
• Affects perception: Creates effects like a straw looking bent in a glass of water.

This is why the fish can see objects above water within a certain angle; it's due to how light refracts at the water's surface before reaching the fish's eyes.

Optics

Optics is a branch of physics dealing with light and its interactions with different materials. This field studies how light behaves, including how it's transmitted, absorbed, and reflected.
Optics encompasses both reflection, like total internal reflection, and refraction, which accounts for changes in light direction.

• Applications: Used in a variety of devices such as glasses, cameras, and telescopes.
• Explores light properties: Looks at light’s wave-like and particle-like properties.
• Fuels innovation: Drives technology advances in laser tech and fiber optics.

Optics not only explains simple everyday phenomena, such as why the sky is blue, but also enables advanced applications like internal reflections, as seen when observing fish underwater.