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 90^o

So, we know that Sin 90^o 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 41.1^o

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 41.1^o .

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 48.75^o ≈ 49^o

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.