Problem 90 - Master Resource Book in JEE Main Physics

Question

Total internal reflection is the phenomenon of reflection of light into denser medium at the interface of denser medium with a rarer medium. Light must. travel from denser to rarer and angle incidence denser medium must be greater than critical angle (C) for the pair of media in contact. We can show that $$ \mu=\frac{1}{\sin C} $$ A point source of light is held at a depth \(h\) below the surface of water. If \(C\) is critical angle of air water interface, the diameter of cirele of light coming from water surface would be (a) \(2 \underline{h} \tan C\) (b) \(h \tan C\) (c) \(h \sin C\) (d) \(h / \sin C\)

Step-by-Step Solution

Answer

The diameter of the circle of light is \(2h \tan C\), corresponding to option (a).

Step 1: Understanding the Situation

A point source of light is submerged at a depth of \( h \) beneath the water surface. Here, we are required to find the diameter of the circular light that reaches the surface of the water, passing through the water-air interface without getting totally internally reflected.

Step 2: Critical Angle and Relation to Refractive Index

For total internal reflection, the critical angle \( C \) is related to the refractive index \( \mu \) by the equation \( \mu = \frac{1}{\sin C} \). As light travels from water (denser) to air (rarer), at \( \theta = C \), the light grazes along the boundary without being reflected internally.

Step 3: Geometry of the Situation

At the critical angle \( C \), light from the source reaches the interface and forms a circular boundary. The radius of this circle can be found using basic trigonometry, where the radius \( r = h \tan C \). The diameter, therefore, is twice the radius.

Step 4: Calculating the Diameter

The diameter \( D \) of the light circle is calculated as: \( D = 2r = 2h \tan C \). This matches with option (a), indicating that the diameter of the circle of light is twice the depth multiplied by the tangent of the critical angle.

Key Concepts

Critical Angle

The concept of the critical angle is crucial in understanding total internal reflection, particularly at the boundary between two different media, such as water and air. When light travels from a denser medium (like water) to a rarer medium (like air), the critical angle is the specific angle of incidence at which the refracted light ray grazes the boundary. At this point, the refracted light travels along the interface rather than passing into the second medium.

It is important to note that:

• The critical angle depends on the refractive indices of the two media.
• For light to be totally internally reflected, the angle of incidence must exceed the critical angle.
• Total internal reflection does not occur if light travels from a rarer to a denser medium.

The mathematical relationship that determines the critical angle is expressed with the formula: \[\mu = \frac{1}{\sin C}\]where \( \mu \) represents the refractive index of the denser medium relative to the rarer medium, and \( C \) is the critical angle.

Refractive Index

Understanding the refractive index is fundamental in optics. The refractive index, symbolized by \( \mu \), is a measure of how much a light ray bends or refracts when it enters a medium from another medium. It is a dimensionless number that provides insight into how light behaves as it passes through different substances.

Key points about the refractive index include:

• A higher refractive index indicates that light slows down more and bends towards the normal when entering the medium.
• The index is dependent on the wavelength of light, which leads to dispersive effects like rainbows in prisms.
• When light travels from a medium with a lower refractive index to one with a higher refractive index, it bends towards the normal.

The refractive index is linked to the critical angle through the equation \( \mu = \frac{1}{\sin C} \), which allows us to calculate either the critical angle or the refractive index if one of these values is known.

Optics

The field of optics is all about the behavior and properties of light. It encompasses various phenomena such as reflection, refraction, and diffraction. A core area in optics is the study of how light interacts with surfaces and passes through different media, following principles that have been established for centuries.

Some essential aspects of optics include:

• Total internal reflection, which occurs when light can't escape a dense medium and is completely reflected back inside.
• Understanding how lenses and mirrors focus or spread light through refraction and reflection, respectively.
• Applications in technology, such as in fiber optic cables where total internal reflection is harnessed to transmit data efficiently.

Optics helps explain natural phenomena like rainbows, the blue color of the sky, and optical illusions. Moreover, it is essential for designing a wide range of optical devices such as telescopes, cameras, and glasses.