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Ray Diagrams

Light is one of the cornerstones of physics. By understanding how light behaves, we have been able to create tools that change the way it travels. Some examples include mirrors and lenses, which we combine to make telescopes for studying distant stars or microscopes to observe life at the microscopic level. We can understand how mirrors and lenses work using…

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Ray Diagrams

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Jetzt kostenlos anmeldenLight is one of the cornerstones of physics. By understanding how light behaves, we have been able to create tools that change the way it travels. Some examples include mirrors and lenses, which we combine to make telescopes for studying distant stars or microscopes to observe life at the microscopic level. We can understand how mirrors and lenses work using ray diagrams. These diagrams show us how light behaves, the path it follows, and how it forms the images we see. Let's have a more detailed look at ray diagrams and some of their applications.

In physics, ray diagrams are a visual representation of the propagation of light. They can help us to understand and visualise multiple situations, such as light being reflected off of a mirror or changing its direction while moving through a lens.

A **ray diagram** is a simplified representation of light to study the trajectory that it follows as it moves from one point to another.

In these diagrams, the initial point often represents the source location, while the ending point represents the observer's position. You can think of a ray diagram as a hand-drawn simulation of how light will move to predict where it will end and how it will form the images we see and their characteristics. Each ray represents a beam of light and they are drawn following certain rules that depend on the geometry and properties of the object that the ray encounters on its way. This may sound complicated but is actually a very practical way to simplify the study of light. A deep understanding of how light interacts with different surfaces and materials is very complex and depends on many variables. With a ray diagram we can focus our attention on the essential details simplifying the study of light rays.

But why do we actually use ray diagrams if they are an oversimplification of how light works? It is fair to admit that we are leaving many details out when we use ray diagrams. However, by abstracting ourselves from the complex picture and leaving aside unnecessary details - for example, how much light is absorbed in mirror and how much is reflected, the material of a lens, how many times and at which points does light actually changes direction in a lens - we can focus on the fundamental elements that are truly necessary to predict the behaviour of light in very specific cases. As simple as they are, rays diagrams can be applied to describe correctly multiple situations: differently shaped mirrors, diverging and converging lenses, and even compound lens systems like microscopes and telescopes.

Ray diagrams can sometimes also be called **light ray diagrams**. All ray diagrams follow two basic rules:

- Every ray must be drawn as a straight line
- Every ray must have an arrowhead pointing in the direction that the light is travelling.

Since ray diagrams can represent multiple situations, more rules are applied depending on the specific case. Let's have a look at a ray diagram for a plane mirror to illustrate this.

The diagram below is a ray diagram showcasing the reflection of light in a plane mirror.

There is one main rule to drawing the ray diagram of a mirror: when light reflects, the **angle of incidence** is always equal to the **angle of reflection**.

In the diagram above, the angle of incidence is marked as \(\theta_i\) and the angle of reflection is marked as \(\theta_r\). These angles are measured with respect to a **normal line**.

The **normal line** is an imaginary line that extends perpendicular to the surface of the mirror, from the point where the light ray is incident to it.

The **angle of incidence **is the angle formed between the incident ray and the normal.

The **angle of reflection **is the angle between the reflected ray and the normal.

We've looked at a regular plain mirror, so now let's look at an array of mirrors that can be very useful, called the L-shaped mirror. Below is an example of an L-shaped mirror:

Notice that light reflects twice causing the ray of light to be reflected back in the same direction that it originally came from. This phenomenon is called **retroreflection*** *and it works regardless of the angle of incidence. Moreover, we can scale up this setup by adding a third mirror perpendicular to these two. The result is a mirror with the shape of the corner of a cube. In this three-dimensional array, any light ray is always reflected directly back to its source.

Why does **retroreflection **work regardless of the angle?

To answer this, let's take an arbitrary angle \( \theta_i \) which is reflected in the vertical mirror:

Both the normal and the horizontal mirror are both perpendicular to the vertical mirror, forming 90-degree angles. We also know that the angle of incidence is the same as the angle of reflection, thus \( \theta_i = \theta_r \). By putting all this information into our diagram, we can identify a right triangle with sizes \( 90^\circ \), \( 90 - \theta_i \), and \( x \), where \( x \) is an unknown angle which we would like to calculate.

Since the sum of all internal angles within a triangle must add up to \( 180^\circ \), we can set up an equation and calculate the value of \( x \).

\( x + 90 - \theta_r + 90 = 180 \)

\( x - \theta_r + 180 = 180 \)

\( x - \theta_r = 0 \)

\( x = \theta_r \)

Therefore, the angle that the reflected ray makes with the horizontal mirror is equal to \( \theta_r \). Knowing this angle allows us to calculate the value for the angle incident on the second mirror.

Now, due to the law of reflection, we know that the angle of reflection must also be \( 90 - \theta_r \), and since the normal line makes a 90-degree angle with the horizontal mirror, the reflected ray must make an angle equal to \( \theta_r \) with the horizontal mirror.

As you can see, the original incident ray and the ray that is bounced back after two reflections are parallel to each other! Furthermore, since we used an arbitrary angle, our result must hold true for all angles.

You may have noticed that the diagram above uses both \( \theta_i \) and \( \theta_r \) to represent two angles of the same size. However, the subindex indicates whether it is the angle of incidence "\( i \)" or the angle of reflection "\( r \)". It is vital that you realise the difference between these two, as it may come up on multiple occasions.

Did you know that in 1969 as part of the Apollo 11 moon mission, astronauts were asked to leave a panel with retroreflectors? This was done so that we could aim a laser from the earth at them and guarantee that light would bounce exactly back in the same direction. Some of the most precise measurements of the distance between the moon and the earth we have to this day were obtained with this method. This is a very impactful experiment, and it can be understood with a relatively simple ray diagram, the very same one we have illustrated above!

Fig. 8. A panel with retroreflectors was deployed to reflect a laser beam back to its source on earth. NASA

A **lens** is any shaped piece of material that redirects light in a specific way.

Usually, lenses are made from glass, but they can also be made from any transparent material, for example, plastic or even ice! But how does this material change the direction of light? This is because of a phenomenon called **refraction**.

**Refraction** is the process where light changes direction when it enters or leaves a new medium, and it takes place because light has a different speed in those mediums.

When the light goes through a water-air interface, it changes its direction. This is why an object looks like it is bent when it is partially submerged in a glass of water. The light coming from the submerged part seems to come from a different position than it really is.

As parallel light rays propagating through the air enter a **convex lens** they get refracted coming together at a single point.

A **convex lens** or **converging** **lens** is thicker in the middle than at the edges and concentrates light rays at a single point called the **principal focus**.

For a regular convex lens, the principal focus will always be along the **principal axis**.

The **principal axis** is an imaginary horizontal line that goes through the geometric centre of a lens.

Fig. 10. Light rays parallel to the principal axis converge at the focus after being refracted by the convex less.

Notice that the light refracts twice. The first time the light refracts is when it moves from the air into the lens, and then once again as they leave the lens. However, when we draw ray diagrams we consider that light rays only refract at one point, and we can use a simpler representation for the lens.

Convex lenses can work in either direction, having two foci. We have labelled them as \(F_1\) and \(F_2\).

Instead of analysing how each light ray would refract, we can get a good understanding of the behaviour of the light rays that go through a convex lens by using the following three special cases:

- A light ray parallel to the principal axis passes through the focus on the other side of the lens after refracting.
- A light ray that goes through the optical centre does not have any deflection.
- A light ray passing through the focus moves parallel to the principal axis after refraction.

A **concave lens** is a **diverging lens **that causes light rays that are parallel to the principal axis to disperse after they have been refracted by the lens.

After refraction, light rays spread out in such a way that makes them look like they are emerging from a single point called the **principal focus** of the lens. A concave lens it is rounded inwards, like a shallow cave in the glass! The following diagram illustrates how light rays passing through a concave lens are dispersed.

As with a convex lens, the light refracts twice, once when entering the lens and once when leaving the lens. However, we can simplify this and create a ray diagram like the one below, which represents the same situation as the image above.

Similarly to the case of concave lenses, we can simplify the interaction of the light with a concave lens by using the following three special cases:

- A light ray after diverging will appear to come from the focus if it is parallel to the principal axis before refracting.
- A ray of light will go through the optical centre without any deviation.
- A ray of light going towards the principal focus will refract and afterwards will move parallel to the principal axis.

If you would like to see a more in-depth explanation of ray diagrams to study how images form, and how lenses can magnify images and correct eyesight problems you can take a look at the article on image formation with lenses.

- A
**ray diagram**or**light ray diagram**is a simplified representation of light to study the trajectory that it follows as it moves from one point to another and interacts with different objects that it may encounter on its way. - In a
**ray**diagram, a ray represents a beam of light. - When drawing a ray diagram, each ray of light must be a straight line.
- In a ray diagram, a convex lens is illustrated by a line segment with an arrowhead pointing outwards on the ends.
- A convex lens is a lens that causes rays of light to converge on a point.
- A concave lens is a lens that causes rays of light to diverge away from each other.
- In a ray diagram, a convex lens is illustrated by a line segment with an arrowhead pointing outwards on each end.
- In a ray diagram, a concave lens is illustrated by a line segment with an arrowhead pointing inwards on each end.

The rules depend on the object that the light will interact. For example, there are 3 main rules for drawing a ray diagram of a convex lens:

- A ray parallel to the principal axis will pass through the focus of a lens.
- A ray that passes through the focus will become parallel to the principal axis.
- A ray passing through the optical centre of a lens will come out the other side without any deviation.

We can distinguish two main important diagrams for concave mirrors:

1. When the object is located farther than the focus of the concave, the image is real, inverted, and reduced in size.

2. When the object is located before the focus of a concave mirror, the image is virtual, upright, and enlarged.

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