Multiple-Concept Example 7 provides helpful background for...

Multiple-Concept Example 7 provides helpful background for this problem. The drawing shows a crystalline quartz slab with a rectangular cross section. A ray of light strikes the slab at an incident angle of $\theta_{1}=34^{\circ},$ enters the quartz, and travels to point $P .$ This slab is surrounded by a fluid with a refractive index $n .$ What is the maximum value of $n$ for which total internal reflection occurs at point $P ?$

Key Concepts

Refractive Index

The refractive index is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. This concept is essential for understanding how light bends or refracts when passing from one material into another and plays a central role in determining the conditions under which total internal reflection can occur.

Snell's Law

Snell's Law is a fundamental law in optics that relates the angles of incidence and refraction to the refractive indices of two different media. It is mathematically expressed as n? sin?? = n? sin??. This law is used to predict the direction of a light ray as it passes between media, making it crucial for analyzing problems involving refraction and total internal reflection.

Critical Angle

The critical angle is the minimum angle of incidence in a denser medium above which total internal reflection occurs when light is incident on an interface with a less dense medium. It is derived from Snell’s Law by setting the angle of refraction to 90°. This concept determines the limit beyond which light no longer refracts into the second medium but is entirely reflected back, which is central to many optical applications.

Total Internal Reflection

Total internal reflection is an optical phenomenon that occurs when a light ray traveling within a denser material strikes the boundary with a less dense medium at an angle greater than the critical angle. Instead of refracting through the boundary, all the light is reflected back into the original medium. This principle is widely used in various optical devices and is key to understanding light behavior in scenarios involving interfaces between materials with different refractive indices.

Transcript

00:02 Okay, my name is jamie, and it looks like that we're given a situation where an incident light is going at the side of this crystalline quartz.

00:28 And this crystalline quartz would have an internal ray where we go from its incidence to point p.

00:41 And our goal for this is to find the maximum n of the fluid when.

00:51 The ray undergoes total internal reflective refraction.

01:01 So that would mean that this angle right here would be 90 degrees.

01:08 Okay.

01:09 So to make it easier for all of us, let us label everything right here.

01:17 So i'll call this theta 1, which was given to us, 34 degrees.

01:26 And theta 2, and this would be theta 3.

01:34 And it's 90 degrees, we'll just call it theta 4.

01:39 So it looks like in this case, this crystalline structure, we'll just name it to have an index of a fraction of ng for glass.

01:55 And ng is given to us for crystalline quartz.

01:59 Ng is equal to 1 .544.

02:07 All right, so we pretty much have all the essentials.

02:10 Since there are two incidences in this case, we would name this incident a and this would be incident b.

02:24 Let's start at incident a right here.

02:34 So as usual, we would use snell's law on this.

02:41 Index of a fraction n times sine 34 degrees.

02:47 This would be where the fluid is is equal to n g times sine of theta 2 okay very good i'll just name this equation 1 now for b we would have n times sine theta 4 is equal to ng sine theta 3 now to find theta 3 now to find theta 3 start with theta 4.

03:35 To go with theta 4, it is given to us, actually, we needed it to be 90 degrees for total internal reflection.

03:46 Now for theta 3, however, we would use the interior angle theorem that this one would be theta 2 because of the total, because of the interior angle theorem of this.

04:04 As you can see over here.

04:10 And since we know that this would be 90 degrees, theta 3 would be 90 degrees minus theta 2.

04:23 And we have theta 3.

04:25 Look at it.

04:27 Now let's plug everything in.

04:32 And sine theta 4 would actually be 90 degrees is equal to n.

04:41 G, sine 90 degrees minus theta 2.

04:50 Let me just circle everything here.

04:53 There we go...