Critical Angle and Total Internal Reflection §

Overview §

The phenomenon of Total Internal Reflection (TIR) is a fascinating aspect of optics and is closely related to the concept of the critical angle. This principle has practical applications in fiber optics, gemstone cutting, and even in natural phenomena like mirages.

Critical Angle §

The critical angle is the angle of incidence beyond which light rays reflect entirely back into the medium, instead of refracting out. This angle is specific to the boundary between two media.

Mathematical Expression §

The critical angle, ${\theta }_c$, can be calculated using Snell’s Law: $n_1\sin ({\theta }_1)=n_2\sin ({\theta }_2)$ Here, $n_1$ and $n_2$ are the refractive indices of the two media, and ${\theta }_1$ and ${\theta }_2$ are the angles of incidence and refraction, respectively. At the critical angle, ${\theta }_2$ equals 90 degrees, leading to the formula: $\sin ({\theta }_c)=\frac{n_2}{n_1}$ where $n_1>n_2$.

Total Internal Reflection §

Total Internal Reflection occurs when the incident light is within a medium with a higher refractive index and strikes the boundary at an angle greater than the critical angle. Under these conditions, all the light is reflected back into the medium, creating a mirror-like effect.

Applications §

1. Fiber Optics: TIR is used to transmit light over long distances with minimal loss.
2. Gemstones: Proper cutting angles maximize TIR to enhance brilliance.
3. Periscopes and Binoculars: TIR enhances image brightness and clarity.

Example: Mirage §

A mirage is a naturally occurring optical illusion caused by TIR. On hot days, the air close to the surface heats up more than the air above, creating a gradient in refractive indices. This causes the light from the sky to undergo TIR, making it appear as if the sky (or water) is on the ground.

---

Test Questions §

1. [Basic] Question: What is the critical angle? Back: The critical angle is the angle of incidence beyond which light rays reflect entirely back into the medium, instead of refracting out.
2. [Basic] Question: Explain how Total Internal Reflection is used in fiber optics. Back: In fiber optics, TIR is used to transmit light over long distances with minimal loss. The light is kept within the optical fibers due to TIR at the boundary, which prevents light from escaping.
3. [Basic] Question: Derive the formula for the critical angle using Snell’s Law. Back: Using Snell’s Law, $n_1\sin ({\theta }_1)=n_2\sin ({\theta }_2)$, set ${\theta }_2$ to 90 degrees for TIR. Rearrange to get $\sin ({\theta }_c)=\frac{n_2}{n_1}$, where ${\theta }_c$ is the critical angle.