Lecture 14: Refraction

Transcription

1 Lecture 14: Refraction We know from experience that there are several transparent substances through which light can travel air, water, and glass are three examples When light passes from one such medium into another, it changes direction this effect is called refraction θ 1 The behavior is described by Snell s Law: sin! 1 = n 2 sin! 2 θ 2 n 2 n is the index of refraction, which is a property of the material

2 Index of Refraction We mentioned the speed of light earlier But it turns out that the speed depends on the medium through which light is traveling fastest in vacuum, slower in any other material The index of refraction is the extent to which light slows down: n = v light c Air is pretty close to vacuum (molecules are far apart) so it has n very close to 1

3 Example: Brewster s Angle Light in air hits a glass block with index of refraction n Some is reflected from the surface, and some is refracted in the glass Find the angle of incidence for which the reflected and refracted beams are perpendicular (this is Brewster s Angle) θ 1 θ 2 Want: 90 o!" o!" 3 = 90 o " 2 +" 3 = 90 o θ 3

4 We know that:! 2 =! 1 nsin! 3 = sin! 1 # sin!! 3 = sin "1 1 & $ % n ' ( So our condition becomes: # sin!! 1 + sin "1 1 & $ % n ' ( = 90o # sin! sin "1 1 & $ % n ' ( = 90o "! 1 sin! 1 n tan! 1 = n! 1 = tan "1 n = sin( 90 o "! ) 1 = cos! 1

5 Wavelength of Refracted Light Going back to the wave picture of light for a moment When a light wave goes from one medium to another, the frequency of the light doesn t change otherwise we d get to a situation where the wave at the boundary is both a peak and a trough! But for any wave, the speed (V), frequency (ν) and wavelength (λ) are related by: V =!" Since we know that the speed of a light wave in a medium is c/n, is must also be true that:! in medium =! in vacuum n

6 Dispersion It turns out that the index of refraction is not quite constant for a material it varies with the wavelength of the light typically, n is larger for shorter wavelengths This means that the angle of refraction will be different for different wavelengths White light is a mixture of all the visible wavelengths So when white light refracts, the various components bend through slightly different angles this effect is called dispersion

7 Examples of dispersion We observe dispersion when white light is passed through a prism: Through this example, we see that our eye interprets different wavelengths as colors! red is the longest wavelength we can see, and purple is the shortest

8 Rainbows Rainbows occur when sunlight is dispersed by rain drops: Note total internal reflection here You can see a rainbow when: it s raining and the sun is shining the sun is low in the sky you are facing away from the sun

9 Total internal reflection Let s assume that a light ray is in a medium with index of refraction, headed toward another medium with index n 2, which is less than θ 1 n 2 The angle of refraction should be given by: sin! 2 = n 2 sin! 1

10 But what if θ 1 is big enough that Then we d need to have sinθ 2 > 1 and that can t happen! So the light can t refract! n 2 sin! 1 > 1 Instead, the light acts as if it hit a wall it reflects inside the material, obeying the usual rule that angle of incidence equals angle of reflection: θ 1 θ 1 n 2

11 This phenomenon is called total internal reflection It happens whenever n 2 sin! 1 > 1 sin! 1 > n 2 The angle at which internal reflection begins to happen is called the critical angle, θ c : sin! c = n 2 Much of the world s telecommunication is done using fiber optics, which are based on total internal reflection

12 Fermat s Principle One of the goals of physics is to explain many phenomena starting from a few fundamental principles Newton s Law of Gravity, for example, explained both the motion of the planets and the falling of apples on Earth Fermat came up with a principle that explained the propagation, reflection and refraction of light: light always travels in the path that minimizes the time it takes to get from point A to point B This clearly is consistent with the fact that light travels in straight lines within a given medium The text shows how Snell s Law arises from Fermat s principle

13 Reflection According to Fermat Let s say that a set of light rays start at point A where will the ones that end up at point B hit the mirror? B (x B, y B ) Mirror at x = 0 A (x A, y A ) d B θ 2 θ 1 d A y m Need to find y m that minimizes total distance traveled d = d A + d B = x 2 A + ( y m! y ) 2 A + x 2 B + ( y B! y ) 2 m

14 To find the minimum, we need to take the derivative with respect to y m and set it equal to 0: dd dy m = y m! y A + x 2 A + ( y m! y ) 2 A y B! y m = 0 x 2 B + ( y B! y ) 2 m y m! y A = x 2 A + ( y m! y ) 2 A y B! y m x 2 B + ( y B! y ) 2 m From the figure, we see that this means: sin! 1 = sin! 2! 1 =! 2 exactly as we expected!