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Reflection and Refraction of Light

Light reflects so that the angle of incidence equals the angle of reflection, both measured from the normal. When light enters a denser medium it slows down and bends towards the normal: $n=\dfrac{\sin i}{\sin r}$ . Going the other way, beyond the critical angle all the light reflects back: total internal reflection, used in optical fibres.

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Section 1 of 14

Learning objectives

Use the law of reflection and describe the image in a plane mirror.
Describe refraction towards and away from the normal.
Use $n=\dfrac{\sin i}{\sin r}$ and $n=\dfrac1{\sin c}$ .
Explain total internal reflection and optical fibres.

Pause and think

A straw in a glass of water looks bent at the surface. Is the straw really bent?

Section 2 of 14

Core idea

Start hereThe bent straw

Put a straw in a glass of water and it looks broken at the surface. The straw is fine; the light from it changes speed as it leaves the water and bends. That bending, refraction, is how lenses, glasses and optical fibres work.

The big idea

Reflection: $i=r$ . Refraction: $n=\frac{\sin i}{\sin r}=\frac cv$ . Total internal reflection beyond the critical angle, $\sin c=\frac1n$ .

1. Reflection

Refraction into glass as the angle of incidence changes

angle of incidence (°) = 10

Light slows in glass and bends towards the normal:

r

is always smaller than

i

. Angles are measured from the normal, not the surface.

Angles measured from the normal.
Plane-mirror image: virtual, upright, same size, as far behind as the object is in front.

2. Refraction

Reflection in a plane mirror

i=r

. The image in a plane mirror is upright, the same size, and as far behind the mirror as the object is in front.

n=\frac{\sin i}{\sin r}=\frac cv

$i$: angle in air.
$r$: angle in the material.
$c$: $3\times10^8$ m/s.
$v$: speed in the material.

Worked exampleInto glass

$i=40^\circ$ , $r=25^\circ$ .

Goal: $n$ ; speed in glass.
Strategy: Both forms of the formula.

$n=\frac{0.643}{0.423}=1.52$
$\sin i/\sin r$ .
$v=\frac{3\times10^8}{1.52}=1.97\times10^8\text{ m/s}$
$v=c/n$ .

Answer

1.52

;

1.97\times10^8

m/s.

✓ CheckBends towards the normal (

r<i

): slower ✓.

3. Total internal reflection

The critical angle

$n=\frac{\sin90^\circ}{\sin c}$
At the critical angle, light leaving the glass skims the surface ( $90^\circ$ in air).
$\sin c=\frac1n$
$\sin90^\circ=1$ .

Above

c

, no light escapes: total internal reflection (only from dense to less dense).

Worked exampleGlass, $n=1.5$

Critical angle; what happens at $45^\circ$ inside?

Goal: $c$ ; behaviour.
Strategy: $\sin c=\frac1n$ .

$c=\sin^{-1}\tfrac1{1.5}=41.8^\circ$
Formula.
$45^\circ>41.8^\circ$
Above the critical angle.

Answer

41.8^\circ

; totally internally reflected (prism periscopes, optical fibres).

✓ CheckLarger

n

gives smaller

c

✓.

Common misconceptionAngle to the surface Always measure from the normal, not the surface.

Pause and think

Why does a straw look bent?

Summary

$i=r$ for mirrors.
$n=\sin i/\sin r=c/v$ .
TIR beyond $c$ .

Refractive index: How much light slows in a material.
Normal: Line perpendicular to the surface.

HistoryIbn al-Haytham's optics. Around

1021

in Cairo, Ibn al-Haytham wrote the Book of Optics, showing by experiment that we see because light enters the eye, not because the eye sends out rays. He studied reflection and refraction carefully and is often called the father of the scientific method.

Fun factDiamonds sparkle because of TIR. Diamond has a refractive index of

2.42

, so its critical angle is only about

24^\circ

. Light entering a well-cut diamond is totally internally reflected many times before escaping out of the top, which is why diamonds sparkle so much more than glass.

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