Critical angle Formula

The critical angle in optics refers to the angle of incidence, beyond which the total internal reflection of light occurs. The trajectory of a ray of light that strikes a medium that has a lower refractive index deviates from the normal trajectory. As a result, the angle of exit of the ray is greater than the angle of incidence. This reflection is called internal reflection. Whenever light travels from a medium with a higher refractive index (n1) to a medium with a lower refractive index (n2), the angle of refraction is greater than the angle of incidence. As a result of the difference in the refractive index, the ray bends towards the surface. So the critical angle is defined as the angle of incidence that provides a 90 degree angle of refraction. Note that the critical angle is an angle of incidence value. For the water-air limit, the critical angle is 48.6 degrees. For the boundary between glass and crown water, the critical angle is 61.0 degrees. The actual value of the critical angle depends on the combination of materials present on each side of the boundary.

Let's consider two different media, half i (incident half) and half r (refractive half). The critical angle is that of θi which gives a value of 90 degrees. If this information is substituted in the Snell's Law equation, a generic equation can be obtained to predict the critical angle.

The critical angle = the inverse function of the sine (refraction index / incident index).

The equation is:

θ_crit = sin^-1(n_r/n_i)

We have:

θ_crit = The critical angle.

n_r = refraction index.

n_i = incident index.

Critical angle Questions:

1)What must be the angle of incidence for there to be total internal reflection of a ray going from water (nw = 1.3) to glass ( ng = 1.52)?

Answer: Given the indices for the means by which the ray passes, we resolve.

θ_crit = sin^-1(n_r/n_i) = sin^-1(1.3/1.52) = 1.064rad.

θ_crit = 1.064rad.

2)A ray of light strikes from a medium (n = 1.67) on a surface of separation with the air (n = 1). It calculates the limit or critical angle.

Answer: Given the indices for the means by which the ray passes, we resolve.

θ_crit = sin^-1(n_r/n_i) = sin^-1(1/1.67) = 1.064rad.

θ_crit = 0.64rad.