1. Home
  2. College physics
  3. Geometric optics
  4. Total internal reflection

25.4 Total internal reflection

<< Chapter < Page
  College physics     Page 1 / 8
Chapter >> Page >
  • Explain the phenomenon of total internal reflection.
  • Describe the workings and uses of fiber optics.
  • Analyze the reason for the sparkle of diamonds.

A good-quality mirror may reflect more than 90% of the light that falls on it, absorbing the rest. But it would be useful to have a mirror that reflects all of the light that falls on it. Interestingly, we can produce total reflection using an aspect of refraction .

Consider what happens when a ray of light strikes the surface between two materials, such as is shown in [link] (a). Part of the light crosses the boundary and is refracted; the rest is reflected. If, as shown in the figure, the index of refraction for the second medium is less than for the first, the ray bends away from the perpendicular. (Since n 1 > n 2 size 12{n rSub { size 8{1} }>n rSub { size 8{2} } } {} , the angle of refraction is greater than the angle of incidence—that is, θ 1 > θ 2 size 12{θ rSub { size 8{1} }>θ rSub { size 8{2} } } {} .) Now imagine what happens as the incident angle is increased. This causes θ 2 size 12{θ rSub { size 8{2} } } {} to increase also. The largest the angle of refraction θ 2 size 12{θ rSub { size 8{2} } } {} can be is 90º size 12{"90"°} {} , as shown in [link] (b).The critical angle     θ c size 12{θ rSub { size 8{c} } } {} for a combination of materials is defined to be the incident angle θ 1 size 12{θ rSub { size 8{1} } } {} that produces an angle of refraction of 90º . That is, θ c is the incident angle for which θ 2 = 90º . If the incident angle θ 1 is greater than the critical angle, as shown in [link] (c), then all of the light is reflected back into medium 1, a condition called total internal reflection .

Critical angle

The incident angle θ 1 size 12{θ rSub { size 8{1} } } {} that produces an angle of refraction of 90º size 12{"90"°} {} is called the critical angle, θ c size 12{θ rSub { size 8{1} } } {} .

In the first figure, an incident ray at an angle theta 1 with a perpendicular line drawn at the point of incidence travels from n1 to n2. The incident ray suffers both refraction and reflection. The angle of refraction is theta 2. In the second figure, as theta 1 is increased, the angle of refraction theta 2 becomes 90 degrees and the angle of reflection corresponding to 90 degrees is theta c. In the third figure, theta c greater than theta i, total internal reflection takes place and instead of refraction, reflection takes place and the light ray travels back into medium n1.
(a) A ray of light crosses a boundary where the speed of light increases and the index of refraction decreases. That is, n 2 < n 1 size 12{n rSub { size 8{2} }<n rSub { size 8{1} } } {} . The ray bends away from the perpendicular. (b) The critical angle θ c size 12{θ rSub { size 8{c} } } {} is the one for which the angle of refraction is . (c) Total internal reflection occurs when the incident angle is greater than the critical angle.

Snell’s law states the relationship between angles and indices of refraction. It is given by

n 1 sin θ 1 = n 2 sin θ 2 . size 12{n rSub { size 8{1} } "sin"θ rSub { size 8{1} } =n rSub { size 8{2} } "sin"θ rSub { size 8{2} } } {}

When the incident angle equals the critical angle ( θ 1 = θ c size 12{θ rSub { size 8{1} } =`θ rSub { size 8{c} } } {} ), the angle of refraction is 90º size 12{"90"°} {} ( θ 2 = 90º size 12{θ rSub { size 8{2} } =`"90" rSup { size 8{0} } } {} ). Noting that sin 90º =1 size 12{"sin"" 90"°"=1"} {} , Snell’s law in this case becomes

n 1 sin θ 1 = n 2 . size 12{n rSub { size 8{1} } "sin"θ rSub { size 8{1} } =n rSub { size 8{2} } } {}

The critical angle θ c size 12{q rSub { size 8{c} } } {} for a given combination of materials is thus

θ c = sin − 1 n 2 / n 1 for size 12{θ rSub { size 8{c} } ="sin" rSup { size 8{ - 1} } left ( {n rSub { size 8{2} } } slash {n rSub { size 8{1} } } right )} {} n 1 > n 2 . size 12{n rSub { size 8{1} }>n rSub { size 8{2} } } {}

Total internal reflection occurs for any incident angle greater than the critical angle θ c size 12{q rSub { size 8{c} } } {} , and it can only occur when the second medium has an index of refraction less than the first. Note the above equation is written for a light ray that travels in medium 1 and reflects from medium 2, as shown in the figure.

How big is the critical angle here?

What is the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air?

Strategy

The index of refraction for polystyrene is found to be 1.49 in [link] , and the index of refraction of air can be taken to be 1.00, as before. Thus, the condition that the second medium (air) has an index of refraction less than the first (plastic) is satisfied, and the equation θ c = sin − 1 n 2 / n 1 size 12{θ rSub { size 8{c} } ="sin" rSup { size 8{ - 1} } left ( {n rSub { size 8{2} } } slash {n rSub { size 8{1} } } right )} {} can be used to find the critical angle θ c size 12{θ rSub { size 8{c} } } {} . Here, then, n 2 = 1 . 00 size 12{n rSub { size 8{2} } =1 "." "00"} {} and n 1 = 1 . 49 size 12{n rSub { size 8{1} } =1 "." "49"} {} .

Solution

The critical angle is given by

θ c = sin − 1 n 2 / n 1 . size 12{θ rSub { size 8{c} } ="sin" rSup { size 8{ - 1} } left ( {n rSub { size 8{2} } } slash {n rSub { size 8{1} } } right )} {}

Substituting the identified values gives

θ c = sin − 1 1 . 00 / 1 . 49 = sin − 1 0 . 671 42.2º. alignl { stack { size 12{θ rSub { size 8{c} } ="sin" rSup { size 8{ - 1} } left ( {1 "." "00"} slash {1 "." "49"} right )="sin" rSup { size 8{ - 1} } left (0 "." "671" right )} {} #="42" "." 2° "." {} } } {}

Discussion

This means that any ray of light inside the plastic that strikes the surface at an angle greater than 42.2º will be totally reflected. This will make the inside surface of the clear plastic a perfect mirror for such rays without any need for the silvering used on common mirrors. Different combinations of materials have different critical angles, but any combination with n 1 > n 2 size 12{n rSub { size 8{1} }>n rSub { size 8{2} } } {} can produce total internal reflection. The same calculation as made here shows that the critical angle for a ray going from water to air is 48 . 6º size 12{"48" "." 6°} {} , while that from diamond to air is 24 . 4º size 12{"24" "." 4°} {} , and that from flint glass to crown glass is 66 . 3º size 12{"66" "." 3°} {} . There is no total reflection for rays going in the other direction—for example, from air to water—since the condition that the second medium must have a smaller index of refraction is not satisfied. A number of interesting applications of total internal reflection follow.

Got questions? Get instant answers now!

Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 4

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask