Given a slab with refractive index n=1.33 and incident light striking the top horizontal face at angle i.The maximum value of i for which total internal reflection occurs is:
n1 sin theta 1 = n2 sin theta 2
1.33 sin theta 1 = 1 sin 90
sin theta = 1/1.33
theta = 48.8 degrees
That is actually the minimum angle for which total internal reflection occurs. Any less than that and it gets through
Total internal reflection occurs when the angle of incidence (i) is greater than the critical angle (θc). The critical angle can be determined using Snell's law:
n1 * sin(i) = n2 * sin(r)
In this case, n1 is the refractive index of the medium the light is coming from (air, with a refractive index of approximately 1), n2 is the refractive index of the medium the light is entering (the slab with a refractive index of 1.33), and r is the angle of refraction.
When total internal reflection occurs, the angle of refraction is 90 degrees (i.e., the light is refracted along the interface between the two media). Therefore, we can rewrite Snell's law as:
n1 * sin(i) = n2 * sin(90)
Simplifying further:
n1 * sin(i) = n2
Now, substituting the values:
1 * sin(i) = 1.33
sin(i) = 1.33
To find the maximum value of i for total internal reflection, we need to find the inverse sine (or arcsine) of 1.33:
i = arcsin(1.33)
Using a calculator, we find that the arcsine of 1.33 is not a real value. This means that there is no angle of incidence for which total internal reflection occurs. The refractive index of the slab is not high enough to cause total internal reflection.
To determine the maximum value of the incident angle (i), we need to consider the phenomenon of total internal reflection.
Total internal reflection occurs when light passes from a material with a higher refractive index to a material with a lower refractive index. In this case, the incident light is passing from the slab (with refractive index n = 1.33) into another medium, likely air with a refractive index of approximately 1.
According to Snell's Law, the relationship between the incident angle (i) and the refracted angle (r) at the interface between two media is given by:
n1 * sin(i) = n2 * sin(r)
Where n1 is the refractive index of the first medium (slab) and n2 is the refractive index of the second medium (air).
For total internal reflection to occur, the refracted angle (r) must be 90 degrees. This means that sin(r) = 1. Substituting this into Snell's Law, we have:
n1 * sin(i) = n2 * 1
Since n2 = 1 for air, the equation becomes:
n1 * sin(i) = 1
Rearranging this equation to solve for sin(i), we have:
sin(i) = 1 / n1
Finally, we can substitute the given value of n1 = 1.33 into the equation to find sin(i):
sin(i) = 1 / 1.33 ≈ 0.7519
To determine the maximum value of i for which total internal reflection occurs, we need to find the inverse sine (sin^(-1)) of 0.7519:
i = sin^(-1)(0.7519)
Using a calculator, we find that i ≈ 49.57 degrees.
Therefore, the maximum value of the incident angle (i) for which total internal reflection occurs is approximately 49.57 degrees.