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You are indoors on a sunny day. A beam of sunlight through a window lights up a section of the floor. How would you draw this sunbeam? You might draw a series of parallel lines showing the path of the sunlight from the window to the floor. This is not exactly accurate – no matter how hard you look, you will not find unique lines of light in the sunbeam! However, this is a good way to draw light. It is also a good way to model light geometrically, as we will see in this chapter.
These narrow, imaginary lines of light are called light rays. Since light is an electromagnetic wave, you could think of a light ray as the path of a point on the crest of a wave. Or, you could think of a light ray as the path taken by a miniscule particle that carries light. We will always draw them the same way: as straight lines between objects, images, and optical devices.
We can use light rays to model mirrors, lenses, telescopes, microscopes, and prisms. The study of how light interacts with materials is optics. When dealing with light rays, we are usually interested in the shape of a material and the angles at which light rays hit it. From these angles, we can work out, for example, the distance between an object and its reflection. We therefore refer to this kind of optics as geometrical optics.
In physics we use the idea of a light ray to indicate the direction that light travels. Light rays are lines with arrows and are used to show the path that light travels. In Figure 1, the light rays from the object enters the eye and the eye sees the object.
The most important thing to remember is that we can only see an object when light from the object enters our eyes. The object must be a source of light (for example a light bulb) or else it must reflect light from a source (for example the moon), and the reflected light enters our eyes.
Light rays are straight lines with arrows to show the path of light.
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Apparatus:
You will need a candle, matches and three sheets of paper.
Method:
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Conclusions:
In the investigation you will notice that the holes in the paper need to be in a straight line. This shows that light travels in a straight line. We cannot see around corners. This also proves that light does not bend around a corner, but travels straight.
On a sunny day, stand outside and look at something in the distance, for example a tree, a flower or a car. From what we have learnt, we can see the tree, flower or car because light from the object is entering our eye. Now take a sheet of paper and hold it about 20 cm in front of your face. Can you still see the tree, flower or car? Why not?
Figure 3 shows that a sheet of paper in front of your eye prevents light rays from reaching your eye.
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Objects cast shadows when light shines on them. This is more evidence that light travels in straight lines. The picture below shows how shadows are formed.
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A ray diagram is a drawing that shows the path of light rays. Light rays are drawn using straight lines and arrow heads. The figure below shows some examples of ray diagrams.
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When you smile into a mirror, you see your own face smiling back at you. This is caused by the reflection of light rays on the mirror. Reflection occurs when a light ray bounces off a surface.
In Transverse Pulses and Transverse Waves we saw that when a pulse or wave strikes a surface it is reflected. This means that waves bounce off things. Sound waves bounce off walls, light waves bounce off mirrors, radar waves bounce off aeroplanes and it can explain how bats can fly at night and avoid things as thin as telephone wires. The phenomenon of reflection is a very important and useful one.
We will use the following terminology. The incoming light ray is called the incident ray. The light ray moving away from the surface is the reflected ray. The most important characteristic of these rays is their angles in relation to the reflecting surface. These angles are measured with respect to the normal of the surface. The normal is an imaginary line perpendicular to the surface. The angle of incidence,
When a ray of light is reflected, the reflected ray lies in the same plane as the incident ray and the normal. This plane is called the plane of incidence and is shown in Figure 8.
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The Law of Reflection states that the angles of incidence and reflection are always equal and that the reflected ray always lies in the plane of incidence.
The Law of Reflection states that the angle of incidence is equal to the angle of reflection.
The simplest example of the law of incidence is if the angle of incidence is
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If the angle of incidence is not
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An incident ray strikes a smooth reflective surface at an angle of 33
We are given the angle between the incident ray and the surface normal. This is the angle of incidence.
We are required to calculate the angle of reflection.
We can use the Law of Reflection, which states that the angle of incidence is equal to the angle of reflection.
We are given the angle of incidence to be 33
The Law of Reflection holds for every light ray. Does this mean that when parallel rays approach a surface, the reflected rays will also be parallel? This depends on the texture of the reflecting surface.
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Figure 11(a) shows a surface that is flat and even. Parallel incident light rays hit the smooth surface and parallel reflected light rays leave the surface. This type of reflection is called specular reflection. Specular reflection occurs when rays are reflected from a smooth, shiny surface. The normal to the surface is the same at every point on the surface. Parallel incident rays become parallel reflected rays. When you look in a mirror, the image you see is formed by specular reflection.
Figure 11(b) shows a surface with bumps and curves. When multiple rays hit this uneven surface, diffuse reflection occurs. The incident rays are parallel but the reflected rays are not. Each point on the surface has a different normal. This means the angle of incidence is different at each point. Then according to the Law of Reflection, each angle of reflection is different. Diffuse reflection occurs when light rays are reflected from bumpy surfaces. You can still see a reflection as long as the surface is not too bumpy. Diffuse reflection enables us to see all objects that are not sources of light.
A bouncing ball can be used to demonstrate the basic difference between specular and diffuse reflection.
Aim:
To demonstrate and compare specular and diffuse reflection.
Apparatus:
You will need:
Method:
Conclusions:
You should have seen that the ball bounces (is reflected off the floor) in a predictable manner off the smooth floor, but bounces unpredictably on the rough ground.
The ball can be seen to be a ray of light and the floor or ground is the reflecting surface. For specular reflection (smooth surface), the ball bounces predictably. For diffuse reflection (rough surface), the ball bounces unpredictably.
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In the previous sections we studied light reflecting off various surfaces. What happens when light passes through a medium? The speed of light, like that of all waves, is dependent on the medium in which it is travelling. When light moves from one medium into another (for example, from air to glass), the speed of light changes. The effect is that the light ray passing into a new medium is refracted, or bent. Refraction is therefore the bending of light as it moves from one optical medium to another.
Refraction is the bending of light that occurs because light travels at different speeds in different materials.
When light travels from one medium to another, it will be bent away from its original path. When it travels from an optically dense medium like water or glass to a less dense medium like air, it will be refracted away from the normal (Figure 15). Whereas, if it travels from a less dense medium to a denser one, it will be refracted towards the normal (Figure 16).
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Just as we defined an angle of reflection in the previous section, we can similarly define an angle of refraction as the angle between the surface normal and the refracted ray. This is shown in Figure 17.
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Which is easier to travel through, air or water? People usually travel faster through air. So does light! The speed of light and therefore the degree of bending of the light depends on the refractive index of material through which the light passes. The refractive index (symbol
The refractive index of a material is the ratio of the speed of light in a vacuum to its speed in the medium.
The symbol
For purposes of calculation, we use
The refractive index (symbol
where
| refractive index (no unit) | |
| speed of light in a vacuum ( |
|
| speed of light in a given medium ( |
Using
we can also examine how the speed of light changes in different media, because the speed of light in a vacuum (
If the refractive index
Table 2 shows refractive indices for various materials. Light travels slower in any material than it does in a vacuum, so all values for
| Medium | Refractive Index |
| Vacuum | 1 |
| Helium | 1,000036 |
| Air* | 1,0002926 |
| Carbon dioxide | 1,00045 |
| Water: Ice | 1,31 |
| Water: Liquid ( |
1,333 |
| Acetone | 1,36 |
| Ethyl Alcohol (Ethanol) | 1,36 |
| Sugar solution (30%) | 1,38 |
| Fused quartz | 1,46 |
| Glycerine | 1,4729 |
| Sugar solution (80%) | 1,49 |
| Rock salt | 1,516 |
| Crown Glass | 1,52 |
| Sodium chloride | 1,54 |
| Polystyrene | 1,55 to 1,59 |
| Bromine | 1,661 |
| Sapphire | 1,77 |
| Glass (typical) | 1,5 to 1,9 |
| Cubic zirconia | 2,15 to 2,18 |
| Diamond | 2,419 |
| Silicon | 4,01 |
Now that we know that the degree of bending, or the angle of refraction, is dependent on the refractive index of a medium, how do we calculate the angle of refraction?
The angles of incidence and refraction when light travels from one medium to another can be calculated using Snell's Law.
where
| Refractive index of material 1 | |
| Refractive index of material 2 | |
| Angle of incidence | |
| Angle of refraction |
Remember that angles of incidence and refraction are measured from the normal, which is an imaginary line perpendicular to the surface.
Suppose we have two media with refractive indices
A light ray with an angle of incidence of 35
From Table 2, the refractive index is 1,333 for water and about 1 for air. We know the angle of incidence, so we are ready to use Snell's Law.
According to Snell's Law:
Since
The light ray passes from a medium of high refractive index to one of low refractive index. Therefore, the light ray is bent away from the normal.
A light ray passes from water to diamond with an angle of incidence of
From Table 2, the refractive index is 1,333 for water and 2,42 for diamond. We know the angle of incidence, so we are ready to use Snell's Law.
According to Snell's Law:
The light ray passes from a medium of low refractive index to one of high refractive index. Therefore, the light ray is bent towards the normal.
If
then from Snell's Law,
For angles smaller than
This means that the angle of incidence is greater than the angle of refraction and the light ray is bent toward the normal.
Similarly, if
then from Snell's Law,
For angles smaller than
This means that the angle of incidence is less than the angle of refraction and the light ray is away toward the normal.
Both these situations can be seen in Figure 18.
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What happens to a ray that lies along the normal line? In this case, the angle of incidence is
This shows that if the light ray is incident at
The angles of incidence and refraction were measured in five unknown media and recorded in the table below. Use your knowledge about Snell's Law to identify each of the unknown media A - E. Use Table 2 to help you.
| Medium 1 |
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Unknown Medium |
| Air | 1,000036 | 38 | 11,6 | ? | A |
| Air | 1,000036 | 65 | 38,4 | ? | B |
| Vacuum | 1 | 44 | 0,419 | ? | C |
| Air | 1,000036 | 15 | 29,3 | ? | D |
| Vacuum | 1 | 20 | 36,9 | ? | E |
Zingi and Tumi performed an investigation to identify an unknown liquid. They shone a beam of light into the unknown liquid, varying the angle of incidence and recording the angle of refraction. Their results are recorded in the following table:
| Angle of Incidence | Angle of Refraction |
| 0,0 |
0,00 |
| 5,0 |
3,76 |
| 10,0 |
7,50 |
| 15,0 |
11,2 |
| 20,0 |
14,9 |
| 25,0 |
18,5 |
| 30,0 |
22,1 |
| 35,0 |
25,5 |
| 40,0 |
28,9 |
| 45,0 |
32,1 |
| 50,0 |
35,2 |
| 55,0 |
38,0 |
| 60,0 |
40,6 |
| 65,0 |
43,0 |
| 70,0 |
? |
| 75,0 |
? |
| 80,0 |
? |
| 85,0 |
? |
| Khan academy video on Snell's Law - 1 |
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Imagine a coin on the bottom of a shallow pool of water. If you reach for the coin, you will miss it because the light rays from the coin are refracted at the water's surface.
Consider a light ray that travels from an underwater object to your eye. The ray is refracted at the water surface and then reaches your eye. Your eye does not know Snell's Law; it assumes light rays travel in straight lines. Your eye therefore sees the image of the at coin shallower location. This shallower location is known as the apparent depth.
The refractive index of a medium can also be expressed as
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A coin is placed at the bottom of a 40 cm deep pond. The refractive index for water is 1,33. How deep does the coin appear to be?
n = 1,33
real depth = 40 cm
apparent depth = ?
The coin appears to be 30,08 cm deep.
A R1 coin appears to be 7 cm deep in a colourless liquid known to be listed in Table 2. The depth of the liquid is 10,43 cm.
real depth = 7 cm
apparent depth = 10,43 cm
Identify the liquid.
Use Table 2. The liquid is an 80% sugar solution.
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A mirror is a highly reflective surface. The most common mirrors are flat and are known as plane mirrors. Household mirrors are plane mirrors. They are made of a flat piece of glass with a thin layer of silver nitrate or aluminium on the back. However, other mirrors are curved and are either convex mirrors or are concave mirrors. The reflecting properties of all three types of mirrors will be discussed in this section.
An image is a representation of an object formed by a mirror or lens. Light from the image is seen.
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If you place a candle in front of a mirror, you now see two candles. The actual, physical candle is called the object and the picture you see in the mirror is called the image. The object is the source of the incident rays. The image is the picture that is formed by the reflected rays.
The object could be an actual source that emits light, such as a light bulb or a candle. More commonly, the object reflects light from another source. When you look at your face in the mirror, your face does not emit light. Instead, light from a light bulb or from the sun reflects off your face and then hits the mirror. However, in working with light rays, it is easiest to pretend the light is coming from the object.
An image formed by reflection may be real or virtual. A real image occurs when light rays actually intersect at the image. A real image is inverted, or upside down. A virtual image occurs when light rays do not actually meet at the image. Instead, you "see" the image because your eye projects light rays backward. You are fooled into seeing an image! A virtual image is erect, or right side up (upright).
You can tell the two types apart by putting a screen at the location of the image. A real image can be formed on the screen because the light rays actually meet there. A virtual image cannot be seen on a screen, since it is not really there.
To describe objects and images, we need to know their locations and their sizes. The distance from the mirror to the object is the object distance,
The distance from the mirror to the image is the image distance,
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When you look into a mirror, you see an image of yourself.
The image created in the mirror has the following properties:
Virtual images are images formed in places where light does not really reach. Light does not really pass through the mirror to create the image; it only appears to an observer as though the light were coming from behind the mirror. Whenever a mirror creates an image which is virtual, the image will always be located behind the mirror where light does not really pass.
A virtual image is upright, on the opposite side of the mirror as the object, and light does not actually reach it.
We draw ray diagrams to predict the image that is formed by a plane mirror. A ray diagram is a geometrical picture that is used for analyzing the images formed by mirrors and lenses. We draw a few characteristic rays from the object to the mirror. We then follow ray-tracing rules to find the path of the rays and locate the image.
The ray diagram for the image formed by a plane mirror is the simplest possible ray diagram. (Reference) shows an object placed in front of a plane mirror. It is convenient to have a central line that runs perpendicular to the mirror. This imaginary line is called the principal axis.
Ray diagrams
The following should be remembered when drawing ray diagrams:
Method: Ray Diagrams for Plane Mirrors
Ray diagrams are used to find the position and size and whether the image is real or virtual.
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Suppose a light ray leaves the top of the object traveling parallel to the principal axis. The ray will hit the mirror at an angle of incidence of 0 degrees. We say that the ray hits the mirror normally. According to the law of reflection, the ray will be reflected at 0 degrees. The ray then bounces back in the same direction. We also project the ray back behind the mirror because this is what your eye does.
Another light ray leaves the top of the object and hits the mirror at its centre. This ray will be reflected at the same angle as its angle of incidence, as shown. If we project the ray backward behind the mirror, it will eventually cross the projection of the first ray we drew. We have found the location of the image! It is a virtual image since it appears in an area that light cannot actually reach (behind the mirror). You can see from the diagram that the image is erect and is the same size as the object. This is exactly as we expected.
We use a dashed line to indicate that the image is virtual.
| Khan academy video on mirrors - 1 |
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The second class of mirrors that we will look at are spherical mirrors. These mirrors are called spherical mirrors because if you take a sphere and cut it as shown in Figure 33 and then polish the inside of one and the outside of the other, you will get a concave mirror and convex mirror as shown. These two mirrors will be studied in detail.
The centre of curvature is the point at the centre of the sphere and describes how big the sphere is.
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The first type of curved mirror we will study are concave mirrors. Concave mirrors have the shape shown in Figure 34. As with a plane mirror, the principal axis is a line that is perpendicular to the centre of the mirror.
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If you think of light reflecting off a concave mirror, you will immediately see that things will look very different compared to a plane mirror. The easiest way to understand what will happen is to draw a ray diagram and work out where the images will form. Once we have done that it is easy to see what properties the image has.
First we need to define a very important characteristic of the mirror. We have seen that the centre of curvature is the centre of the sphere from which the mirror is cut. We then define that a distance that is half-way between the centre of curvature and the mirror on the principal axis. This point is known as the focal point and the distance from the focal point to the mirror is known as the focal length (symbol
The focal point of a mirror is the midpoint of a line segment joining the vertex and the centre of curvature. It is the position at which all parallel rays are focussed.
Why are we making such a big deal about this point we call the focal point? It has an important property we will use often. A ray parallel to the principal axis hitting the mirror will always be reflected through the focal point. The focal point is the position at which all parallel rays are focussed.
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From Figure 36, we see that the image created by a concave mirror is real and inverted, as compared to the virtual and erect image created by a plane mirror.
A real image can be cast on a screen; it is inverted, and on the same side of the mirror as the object.
A concave mirror is also known as a converging mirror. Light rays appear to converge to the focal point of a concave mirror.
The second type of curved mirror we will study are convex mirrors. Convex mirrors have the shape shown in Figure 37. As with a plane mirror, the principal axis is a line that is perpendicular to the centre of the mirror.
We have defined the focal point as that point that is half-way along the principal axis between the centre of curvature and the mirror. Now for a convex mirror, this point is behind the mirror. A convex mirror has a negative focal length because the focal point is behind the mirror.
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To determine what the image from a convex mirror looks like and where the image is located, we need to remember that a mirror obeys the laws of reflection and that light appears to come from the image. The image created by a convex mirror is shown in Figure 38.
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From Figure 38, we see that the image created by a convex mirror is virtual and upright, as compared to the real and inverted image created by a concave mirror.
A convex mirror is also known as a diverging mirror. Light rays appear to diverge from the focal point of a convex mirror.
The properties of mirrors are summarised in Table 6.
| Plane | Concave | Convex |
| – | converging | diverging |
| virtual image | real image | virtual image |
| upright | inverted | upright |
| image behind mirror | image in front of mirror | image behind mirror |
In Figure 36 and Figure 38, the height of the object and image arrows were different. In any optical system where images are formed from objects, the ratio of the image height,
This is true for the mirror examples we showed above and will also be true for lenses, which will be introduced in the next sections. For a plane mirror, the height of the image is the same as the height of the object, so the magnification is simply
A concave mirror forms an image that is 4,8 cm high. The height of the object is 1,6 cm. Calculate the magnification of the mirror.
Image height
Object height
Magnification
The magnification is 3 times.
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Work in groups of four. Each group will need a raybox (or torch) with slit, triangular glass prism and protractor. If you do not have a raybox, use a torch and stick two pieces of tape over the lens so that only a thin beam of light is visible.
Aim:
To investigate total internal reflection.
Method:
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When we increase the angle of incidence, we reach a point where the angle of refraction is 90
The critical angle is the angle of incidence where the angle of reflection is 90
If the angle of incidence is bigger than this critical angle, the refracted ray will not emerge from the medium, but will be reflected back into the medium. This is called total internal reflection.
Total internal reflection takes place when
Total internal reflection takes place when light is reflected back into the medium because the angle of incidence is greater than the critical angle.
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Each medium has its own unique critical angle. For example, the critical angle for glass is 42
Now we shall learn how to derive the value of the critical angle for two given media. The process is fairly simple and involves just the use of Snell's Law that we have already studied. To recap, Snell's Law states:
where
However, we also know that the angle of refraction at the critical angle is 90
We can then write Snell's Law as:
Solving for
| Khan academy video on refraction - 1 |
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Given that the refractive indices of air and water are 1 and 1,33, respectively, find the critical angle.
We know that the critical angle is given by:
The critical angle for light travelling from water to air is
Complete the following ray diagrams to show the path of light in each situation.
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The critical angle for water is
We are asked to complete the diagrams.
For incident angles smaller than
For incident angles greater than
For incident angles equal to
The light must travel from a high optical density to a lower one.
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Refraction occurs (ray is bent away from the normal)
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Total internal reflection occurs
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Refraction towards the normal (air is less dense than water)
Total internal reflection is a powerful tool since it can be used to confine light. One of the most common applications of total internal reflection is in fibre optics. An optical fibre is a thin, transparent fibre, usually made of glass or plastic, for transmitting light. Optical fibres are usually thinner than a human hair! The construction of a single optical fibre is shown in Figure 55.
The basic functional structure of an optical fibre consists of an outer protective cladding and an inner core through which light pulses travel. The overall diameter of the fibre is about 125
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Optical fibres are most common in telecommunications, because information can be transported over long distances, with minimal loss of data. The minimised loss of data gives optical fibres an advantage over conventional cables.
Data is transmitted from one end of the fibre to another in the form of laser pulses. A single strand is capable of handling over 3000 simultaneous transmissions which is a huge improvement over the conventional co-axial cables. Multiple signal transmission is achieved by sending individual light pulses at slightly different angles. For example if one of the pulses makes a 72,23
Optic fibres are used in medicine in endoscopes.
Endoscopy means to look inside and refers to looking inside the human body for diagnosing medical conditions.
The main part of an endoscope is the optical fibre. Light is shone down the optical fibre and a medical doctor can use the endoscope to look inside a patient. Endoscopes are used to examine the inside of a patient's stomach, by inserting the endoscope down the patient's throat.
Endoscopes allow minimally invasive surgery. This means that a person can be diagnosed and treated through a small incision. This has advantages over open surgery because endoscopy is quicker and cheaper and the patient recovers more quickly. The alternative is open surgery which is expensive, requires more time and is more traumatic for the patient.
| Column A | Column B | |
| (a) | Real image | Upright |
| (b) | Virtual image | Can be cast on a screen |
| (c) | Concave mirror | In front |
| (d) | Convex mirror | Behind |
| (e) | Plane mirror | Inverted |
| Light travels to it | ||
| Upside down | ||
| Light does not reach it | ||
| Erect | ||
| Same size |
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This work is licensed by Rory Adams, Free High School Science Texts Project, and Heather Williams under a Creative Commons Attribution License (CC-BY 3.0), and is an Open Educational Resource.
Last edited by Free High School Science Texts Project on Apr 8, 2011 10:22 am GMT-5.
