How do we calculate speed of light?

 It is easy to notice some odd things when looking into a fish tank. For example, you may see the same fish appearing to be in two different places (Figure 

25.3.1$\mathrm{25.3.}1$). This is because light coming from the fish to us changes direction when it leaves the tank, and in this case, it can travel two different paths to get to our eyes. The changing of a light 

ray

’s direction (loosely called bending) when it passes through variations in matter is called 

refraction

. 

Refraction

 is responsible for a tremendous 

range

 of optical phenomena, from the action of lenses to voice transmission through optical fibers.

Definition: 

REFRACTION

The changing of a light 

ray

’s direction (loosely called bending) when it passes through variations in matter is called 

refraction

.

SPEED OF LIGHT

The 

speed of light

 c$�$ not only affects 

refraction

, it is one of the central concepts of Einstein’s 

theory

 of 

relativity

. As the 

accuracy

 of the measurements of the 

speed of light

 were improved, c$�$ was found not to depend on the 

velocity

 of the source or the observer. However, the 

speed of light

 does vary in a precise manner with the material it traverses. These facts have far-reaching implications, as we will see in "

Special Relativity

." It makes connections between space and 

time

 and alters our expectations that all observers measure the same 

time

 for the same event, for example. The 

speed of light

 is so important that its value in a vacuum is one of the most 

fundamental

 constants in nature as well as being one of the four 

fundamental

 

SI units

.

Figure 25.3.1$\mathrm{25.3.}1$: Looking at the fish tank as shown, we can see the same fish in two different locations, because light changes directions when it passes from water to air. In this case, the light can reach the observer by two different paths, and so the fish seems to be in two different places. This bending of light is called refraction and is responsible for many optical phenomena.

Why does light change direction when passing from one material (medium) to another? It is because light changes speed when going from one material to another. So before we study the 

law

 of 

refraction

, it is useful to discuss the 

speed of light

 and how it varies in different media.

The 

Speed of Light

Early attempts to measure the 

speed of light

, such as those made by Galileo, determined that light moved extremely fast, perhaps instantaneously. The first real evidence that light traveled at a finite speed came from the Danish astronomer Ole Roemer in the late 17th century. Roemer had noted that the average orbital 

period

 of one of Jupiter’s moons, as measured from Earth, varied depending on whether Earth was moving toward or away from Jupiter. He correctly concluded that the apparent change in 

period

 was due to the change in 

distance

 between Earth and Jupiter and the 

time

 it took light to travel this 

distance

. From his 1676 data, a value of the 

speed of light

 was calculated to be 2.26×108m/s$2.26\times {10}^8�/�$ (only 25% different from today’s accepted value). In more recent times, physicists have measured the 

speed of light

 in numerous ways and with increasing 

accuracy

. One particularly direct method, used in 1887 by the American physicist Albert Michelson (1852–1931), is illustrated in Figure 25.3.2$\mathrm{25.3.}2$. Light reflected from a rotating set of mirrors was reflected from a stationary 

mirror

 35 km away and returned to the rotating mirrors. The 

time

 for the light to travel can be determined by how fast the mirrors must rotate for the light to be returned to the observer’s eye.

Figure 25.3.2$\mathrm{25.3.}2$: A schematic of early apparatus used by Michelson and others to determine the 

speed of light

. As the mirrors rotate, the reflected 

ray

 is only briefly directed at the stationary 

mirror

. The returning 

ray

 will be reflected into the observer's eye only if the next 

mirror

 has rotated into the correct 

position

 just as the 

ray

 returns. By measuring the correct rotation rate, the 

time

 for the round trip can be measured and the 

speed of light

 calculated. Michelson’s calculated value of the 

speed of light

 was only 0.04% different from the value used today.

The 

speed of light

 is now known to great 

precision

. In fact, the 

speed of light

 in a vacuum c$�$ is so important that it is accepted as one of the basic physical quantities and has the fixed value.

VALUE OF THE 

SPEED OF LIGHT

$$\begin{array}{}\text{(25.3.1)}& �& \equiv 2.99792458\times {10}^8\text{(25.3.2)}& & \sim 3.00\times {10}^8�/�\end{array}$$

The approximate value of 3.00×108m/s$3.00\times {10}^8�/�$ is used whenever three-digit 

accuracy

 is sufficient. The 

speed of light

 through matter is less than it is in a vacuum, because light interacts with atoms in a material. The 

speed of light

 depends strongly on the type of material, since its interaction with different atoms, crystal lattices, and other substructures varies.

Definition: 

INDEX OF REFRACTION

We define the 

index of refraction

 n$�$ of a material to be

n=cv,(25.3.3)$$\begin{array}{}\text{(25.3.3)}& �=\frac{�}{�},\end{array}$$

where v$�$ is the observed 

speed of light

 in the material. Since the 

speed of light

 is always less than c$�$ in matter and equals c$�$ only in a vacuum, the 

index of refraction

 is always greater than or equal to one. That is, n>1$�>1$.

Table 25.3.1$\mathrm{25.3.}1$ gives the indices of 

refraction

 for some representative substances. The values are listed for a particular 

wavelength

 of light, because they vary slightly with 

wavelength

. (This can have important effects, such as colors produced by a prism.) 

Note

 that for gases, n$�$ is close to 1.0. This seems reasonable, since atoms in gases are widely separated and light travels at c$�$ in the vacuum between atoms. It is common to take n=1$�=1$ for gases unless great 

precision

 is needed. Although the 

speed of light

 v$�$ in a medium varies considerably from its value c$�$ in a vacuum, it is still a large speed.

Table 25.3.1$\mathrm{25.3.}1$: 

Index of Refraction

 in Various Media

Medium	n
Gases at 0ºC,1atm$\mathbf{0}º\mathit{�}\mathbf{,}\mathbf{1}\mathit{�}\mathit{�}\mathit{�}$
Air	1.000293
Carbon dioxide	1.00045
Hydrogen	1.000139
Oxygen	1.000271
Liquids at 20ºC
Benzene	1.501
Carbon disulfide	1.628
Carbon tetrachloride	1.461
Ethanol	1.361
Glycerine	1.473
Water, fresh	1.333
Solids at 20ºC
Diamond	2.419
Fluorite	1.434
Glass, crown	1.52
Glass, flint	1.66
Ice at 20ºC	1.309
Polystyrene	1.49
Plexiglas	1.51
Quartz, crystalline	1.544
Quartz, fused	1.458
Sodium chloride	1.544
Zircon	1.923

Example 25.3.1$\mathrm{25.3.}1$: 

Speed of Light

 in Matter

Calculate the 

speed of light

 in 

zircon

, a material used in jewelry to imitate diamond.

Strategy:

The 

speed of light

 in a material, v$�$, can be calculated from the 

index of refraction

 n$�$ of the material using the equation n=c/v$�=�/�$.

Solution

The equation for 

index of refraction

 (Equation 25.3.3$\text{25.3.3}$) can be rearranged to determine v$�$

v=cn.$$�=\frac{�}{�}.$$

The 

index of refraction

 for 

zircon

 is given as 1.923 in Table 25.3.1$\mathrm{25.3.}1$, and c$�$ is given in the equation for 

speed of light

. Entering these values in the last expression gives

$$\begin{array}{rl}�& =\frac{3.00\times {10}^8�/�}{1.923}\\ & =1.56\times {10}^8�/�.\end{array}$$

Discussion:

This speed is slightly larger than half the 

speed of light

 in a vacuum and is still high compared with speeds we normally experience. The only substance listed in Table 25.3.1$\mathrm{25.3.}1$ that has a greater 

index of refraction

 than 

zircon

 is diamond. We shall see later that the large 

index of refraction

 for 

zircon

 makes it sparkle more than glass, but less than diamond.

Law

 of 

Refraction

Figure 25.3.3$\mathrm{25.3.}3$ shows how a 

ray

 of light changes direction when it passes from one medium to another. As before, the angles are measured relative to a perpendicular to the surface at the point where the light 

ray

 crosses it. (Some of the incident light will be reflected from the surface, but for now we will concentrate on the light that is transmitted.) The change in direction of the light 

ray

 depends on how the 

speed of light

 changes. The change in the 

speed of light

 is related to the indices of 

refraction

 of the media involved. In the situations shown in Figure 25.3.3$\mathrm{25.3.}3$, medium 2 has a greater 

index of refraction

 than medium 1. This means that the 

speed of light

 is less in medium 2 than in medium 1. 

Note

 that as shown in Figure 25.3.3a$\mathrm{25.3.}3�$, the direction of the 

ray

 moves closer to the perpendicular when it slows 

down

. Conversely, as shown in Figure 25.3.3b$\mathrm{25.3.}3�$, the direction of the 

ray

 moves away from the perpendicular when it speeds 

up

. The path is exactly reversible. In both cases, you can imagine what happens by thinking about pushing a lawn mower from a footpath onto grass, and vice versa. Going from the footpath to grass, the front wheels are slowed and pulled to the side as shown. This is the same change in direction as for light when it goes from a fast medium to a slow one. When going from the grass to the footpath, the front wheels can move faster and the mower changes direction as shown. This, too, is the same change in direction as for light going from slow to fast.

Figure 25.3.3$\mathrm{25.3.}3$: The change in direction of a light 

ray

 depends on how the 

speed of light

 changes when it crosses from one medium to another. The 

speed of light

 is greater in medium 1 than in medium 2 in the situations shown here. (a) A 

ray

 of light moves closer to the perpendicular when it slows 

down

. This is analogous to what happens when a lawn mower goes from a footpath to grass. (b) A 

ray

 of light moves away from the perpendicular when it speeds 

up

. This is analogous to what happens when a lawn mower goes from grass to footpath. The paths are exactly reversible.

The amount that a light 

ray

 changes its direction depends both on the incident angle and the amount that the speed changes. For a 

ray

 at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in angle. The exact mathematical relationship is the 

law

 of 

refraction

, or "Snell's 

Law

," which is stated in equation form as

THE 

LAW

 OF 

REFRACTION

 (Snell's 

Law

)

n1sinθ1=n2sinθ2.$$\begin{array}{}\text{(25.3.4)}& {�}_1\sin {�}_1={�}_2\sin {�}_2.\end{array}$$

Here, n1${�}_1$ and n2${�}_2$ are the indices of 

refraction

 for medium 1 and 2, and θ1${�}_1$ and θ2${�}_2$ are the angles between the rays and the perpendicular in medium 1 and 2, as shown in Figure 25.3.3$\mathrm{25.3.}3$. The incoming 

ray

 is called the incident 

ray

 and the outgoing 

ray

 the refracted 

ray

, and the associated angles the incident angle and the refracted angle. The 

law

 of 

refraction

 is also called Snell’s 

law

 after the Dutch mathematician Willebrord Snell (1591–1626), who discovered it in 1621. Snell’s experiments showed that the 

law

 of 

refraction

 was obeyed and that a characteristic 

index of refraction

 n$�$ could be assigned to a given medium. Snell was not aware that the 

speed of light

 varied in different media, but through experiments he was able to determine indices of 

refraction

 from the way light rays changed direction.

TAKE-HOME EXPERIMENT: A BROKEN PENCIL

A classic observation of 

refraction

 occurs when a pencil is placed in a glass half filled with water. Do this and observe the shape of the pencil when you look at the pencil sideways, that is, through air, glass, water. Explain your observations. Draw 

ray

 diagrams for the situation.

Example 25.3.2$\mathrm{25.3.}2$: Determine the 

Index of Refraction

 from 

Refraction

 Data

Find the 

index of refraction

 for medium 2 in Figure 25.3.3a$\mathrm{25.3.}3�$, assuming medium 1 is air and given the incident angle is 30.0∘${30.0}^{\circ }$ and the angle of 

refraction

 is 22.0∘${22.0}^{\circ }$.

Strategy

The 

index of refraction

 for air is taken to be 1 in most cases (and 

up

 to four 

significant figures

, it is 1.000). Thus n1=1.00${�}_1=1.00$ here. From the given information, θ1=30.0∘${�}_1={30.0}^{\circ }$ and θ2=22.0∘${�}_2={22.0}^{\circ }$ With this information, the only unknown in Snell’s 

law

 is n2${�}_2$, so that it can be used to find this unknown.

Solution

Snell's 

law

 (Equation 25.3.4$\text{25.3.4}$) can be rearranged to isolate n2${�}_2$ gives

n2=n1sinθ1sinθ2.(25.3.5)$$\begin{array}{}\text{(25.3.5)}& {�}_2={�}_1\frac{\sin {�}_1}{\sin {�}_2}.\end{array}$$

Entering known values,

$$\begin{array}{rl}{�}_2& ={�}_1\frac{\sin {30.0}^{\circ }}{\sin {22.0}^{\circ }}\\ & =\frac{0.500}{0.375}\\ & =\mathrm{1.33.}\end{array}$$

Discussion

This is the 

index of refraction

 for water, and Snell could have determined it by measuring the angles and performing this calculation. He would then have found 1.33 to be the appropriate 

index of refraction

 for water in all other situations, such as when a 

ray

 passes from water to glass. Today we can verify that the 

index of refraction

 is related to the 

speed of light

 in a medium by measuring that speed directly.

Example 25.3.3$\mathrm{25.3.}3$: A Larger Change in Direction

Suppose that in a situation like that in the previous example, light goes from air to diamond and that the incident angle is 30.0∘${30.0}^{\circ }$. Calculate the angle of 

refraction

 θ2${�}_2$ in the diamond.

Strategy

Again the 

index of refraction

 for air is taken to be n1=1.00${�}_1=1.00$, and we are given θ1=30.0∘${�}_1={30.0}^{\circ }$. We can look 

up

 the 

index of refraction

 for diamond in Table 25.3.1$\mathrm{25.3.}1$, finding n2=2.419${�}_2=2.419$. The only unknown in Snell’s 

law

 is θ2${�}_2$, which we wish to determine.

Solution

Solving Snell’s 

law

 (Equation 25.3.4$\text{25.3.4}$) for sinθ2$\sin {�}_2$ yields

sinθ2=n1n2sinθ1.$$\begin{array}{}\text{(25.3.6)}& \sin {�}_2=\frac{{�}_1}{{�}_2}\sin {�}_1.\end{array}$$

Entering known values,

sinθ2$$\begin{array}{rl}\sin {�}_2& =\frac{1.00}{2.419}\sin {30.0}^{\circ }\\ & =\left(0.413\right)\left(0.500\right)\\ & =\mathrm{0.207.}\end{array}$$

The angle is thus

θ2=sin0.207−1=11.9∘.$$\begin{array}{}\text{(25.3.7)}& {�}_2=\sin {0.207}^{-1}={11.9}^{\circ }.\end{array}$$

Discussion

For the same 30∘${30}^{\circ }$ angle of incidence, the angle of 

refraction

 in diamond is significantly smaller than in water (11.9∘${11.9}^{\circ }$ rather than 22∘${22}^{\circ }$ -- see the preceding example).

Summary

• The changing of a light 
  ray
  ’s direction when it passes through variations in matter is called 
  refraction
  .
• The 
  speed of light
   in vacuuum c=2.99792458×108∼3.00×108m/s$�=2.99792458\times {10}^8\sim 3.00\times {10}^8�/�$
• Index of refraction
   n=cv$�=\frac{�}{�}$, where v$�$ is the 
  speed of light
   in the material, c$�$ is the 
  speed of light
   in vacuum, and n$�$ is the 
  index of refraction
  .
• Snell’s 
  law
  , the 
  law
   of 
  refraction
  , is stated in equation form as n1sinθ1=n2sinθ2${�}_1{\sin }_{{�}_1}={�}_2{\sin }_{{�}_2}$.