Step-by-Step Guide to Calculating the Critical Angle
We all are familiar with the concept of the angle of incidence and reflection and refraction. When light strikes the mirror, it reciprocates; however, when the same light is incident at a medium, it gets deviated from its original path; this is actually refraction.
So, the critical angle in Physics is inclusive of incidence, reflection, and refraction. The critical angle is the angle of incidence that produces a 90 degrees angle of refraction.
This page describes the critical angle definition, critical angle formula, and the critical angle and total internal reflection by considering real-life situations.
Define Critical Angle
The critical angle definition explains a lot about the laws of refraction and how the angle of incidence can be adjusted to bring a 90-degree angle of refraction. Here, to understand the formula of critical angle, we will introduce the concept of snell’s law as well. So let’s get started.
You already know that Snell’s law is the law of refraction. We also call it the Snell-Descartes law. It is defined in the following manner:
The formula of critical angle describes the relationship between the angles of incidence and refraction, when considering light or other wave crossing a boundary between two different isotropic media, viz: water, glass, or air.
If the light coming from the air medium passes through the jar filled with water, then we observe that there is a difference in the refractive indices of each.
If n1 is the refractive index of air and Sin θ1 is the angle of incidence when light passes through the air, it suffers refraction in the form of the angle of deviation, i.e., Sin θ2 with water having a refractive index of n2, then we have the formula for Snell’s law as;
n1 Sin θ1 = n2 Sin θ2 ….(1)
Here,
Sin θ1 = angle of incidence for the incident medium
&
Sin θ2 = angle of refraction or the angle of deviation at the refractive medium
So, now, we will derive the critical angle formula from equation (1):
Critical Angle Formula
According to the Critical Angle definition, the incident angle is the critical angle that adjusts the refractive angle to 90-degrees, so in equation (1), we will do the same; let’s do it:
n1 Sin θcritical = n2 Sin 90o
So, we know that Sin 90o is equal to 1 and we get our new equation as;
n1 Sin θcritical = n2
Now, we get the following equation;
n2/n1 = Sin θcritical
According to the inverse trigonometric, we rewrite the above equation in the following manner:
θcritical = Sin-1 (n2/n1) …..(2)
So, equation (2) is the required Critical Angle Formula.
Critical Angle Physics
In this heading, we will discuss the following:
Critical angle of glass
Critical angle of diamond
Critical angle of water
Critical Angle of Glass
We know that the refractive index glass, n1= 1.52
And, the refractive index of air, n2 = 1.00
Using equation (2), and applying these two values as;
θcritical = Sin-1 (1.00/1.52)
On solving, we get the critical angle of glass as 41.1o
Critical Angle of Diamond
We know that the refractive index diamond, n1= 2.42
And, the refractive index of air, n2 = 1.00
Using equation (2), and applying these two values as;
θcritical = Sin-1 (1.00/1.52)
On solving, we get the critical angle of water as 41.1o .
Critical Angle of Water
We know that the refractive index of water, n1 = 1.00
And, the refractive index of air, n2= 1.33
Using equation (2), and applying these two values as;
θcritical = Sin-1 (1.00/1.33)
On solving, we get the critical angle of water to be 48.75o ≈ 49o
Now, the critical angle has a close relationship with the total internal reflection; let’s understand how.
Critical Angle and Total Internal Reflection
We all must have experienced a mirage (a false reflection of the river) on a hot sunny day especially if we are in a desert; this mirage is actually a total internal reflection.
As we know that when the light incidences on the boundary or any medium, it suffers refraction. Now, as this angle of incidence approaches a required limit, which is the critical angle, the angle of refraction becomes 90°, at which the refracted ray of light of an electromagnetic wave becomes parallel to the surface.
Let’s suppose that the angle of incidence increases beyond the critical angle, at this point, the condition of refraction no longer satisfies; so there is no refracted ray, and the partial reflection becomes total. This is how the critical angle and total internal reflection works in real-life.
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Do You Know?
For visible light, the critical angle is approximately 49° for when it incidences at the water-to-air boundary and about 42° when it incidences the common glass-to-air boundary.
FAQs on What Is Critical Angle in Physics?
1. What is the critical angle in Physics?
The critical angle is the specific angle of incidence in a denser medium for which the angle of refraction in the rarer medium is exactly 90 degrees. When light travels from a denser medium (like water or glass) to a rarer medium (like air), it bends away from the normal. The critical angle, denoted as 'c', represents the tipping point beyond which light cannot exit the denser medium.
2. What is the formula used to calculate the critical angle?
The formula to calculate the critical angle (c) is derived from Snell's Law. It is given by: sin(c) = n₂ / n₁, where 'n₁' is the refractive index of the denser medium and 'n₂' is the refractive index of the rarer medium. For light travelling from a medium into air or a vacuum (where n₂ ≈ 1), the formula simplifies to sin(c) = 1 / n₁.
3. What is the relationship between the critical angle and total internal reflection (TIR)?
The critical angle is the essential prerequisite for the phenomenon of Total Internal Reflection (TIR). When the angle of incidence in the denser medium is exactly equal to the critical angle, the refracted ray grazes the boundary surface. If the angle of incidence exceeds the critical angle, the light ray does not refract into the rarer medium at all. Instead, it is completely reflected back into the denser medium. This complete reflection is known as TIR.
4. What are the two essential conditions for total internal reflection to occur?
For total internal reflection to happen, two conditions must be met:
- The light ray must be travelling from a medium with a higher refractive index to a medium with a lower refractive index (i.e., from a denser to a rarer medium).
- The angle of incidence in the denser medium must be greater than the critical angle for that pair of media.
5. Does the critical angle have a fixed value for all materials?
No, the critical angle is not a universal constant. Its value is specific to the pair of media involved and depends entirely on their refractive indices. For example, the critical angle for a water-air interface is approximately 48.8°, while for a crown glass-air interface, it is about 42°. This is because water and glass have different refractive indices.
6. What are the critical angles for common materials like glass and water when light passes into the air?
The critical angle depends on the material's refractive index relative to the surrounding medium (usually air). Here are some common examples:
- For light travelling from water to air, the critical angle is approximately 48.8 degrees.
- For light travelling from common crown glass to air, it is about 42 degrees.
- For light travelling from a diamond to air, the critical angle is much smaller, at only 24.4 degrees.
7. How is the concept of critical angle applied in optical fibres?
The functioning of optical fibres is a prime application of TIR, which relies on the critical angle. An optical fibre consists of a core (high refractive index) and cladding (lower refractive index). Light signals are launched into the core at an angle greater than the critical angle of the core-cladding interface. This causes the light to undergo repeated total internal reflection, allowing it to propagate along the fibre over long distances with minimal energy loss.
8. Why does a diamond sparkle so much? Explain using the concept of critical angle.
A diamond's brilliant sparkle is due to its very high refractive index (about 2.42), which gives it a very low critical angle of approximately 24.4°. When light enters a properly cut diamond, it is very likely to strike an internal facet at an angle greater than this small critical angle. This traps the light, causing it to undergo multiple instances of total internal reflection before it finally exits. This extended path and multiple reflections create the intense, colourful sparkle that diamonds are famous for.
9. What is Snell's Window and how does it relate to the critical angle?
Snell's Window is the phenomenon where an underwater observer sees the entire 180° view of the world above the surface compressed into a circular cone of light. The angle of this cone is determined by the critical angle of water (approx. 48.8°). Light rays from the horizon reach the observer at the critical angle, forming the edge of this window. Any view outside this cone is not of the sky but a mirror-like reflection of the underwater surroundings due to total internal reflection.
