Lesson Explainer: Drawing Ray Diagrams for Concave Mirrors Science • Third Year of Preparatory School

In this explainer, we will learn how to draw diagrams of light rays interacting with concave mirrors.

Before starting to draw ray diagrams, it will be useful to first consider a concave mirror as a three-dimensional solid object.

A concave mirror is a hollow curved object, like a bowl.

A concave mirror is shown in the following figure. The optical axis of the mirror is shown.

The optical axis of a concave mirror is an imaginary line that passes through the point at the back of the mirror.

The optical axis of a spherical mirror is equidistant from the surface of the mirror in every direction perpendicular to this axis. This means that the red and blue lines shown in the following figure are actually the same length.

If the mirror is viewed along the optical axis, it is clearer to see that the mirror is symmetrical around this axis.

An incident light ray can travel along the optical axis of the mirror. This is shown in the following figure.

We can see that an incident light ray that travels along the optical axis of the mirror will hit the point at the back of the mirror and be reflected back along its incoming path.

For rays with paths that are parallel to the optical axis, only rays that travel along the optical axis of a concave mirror are reflected back along their incoming path. Any other path that such a ray travels along results in the ray being reflected on a path different from its incoming path.

The following figure shows how we can represent a cross section of a concave mirror with a curve in a two-dimensional drawing.

The following figure shows light rays incident on a curve representing a cross section of a concave mirror.

Of the three incident light rays shown, only the ray shown in green will reflect along its incoming path. The two rays shown in black will reflect along different paths from their incoming paths.

The following figure shows a magnified view of the part of the mirror where one of the rays shown in black hits the mirror.

We see that where the ray hits the mirror, there is a line normal to the surface of that point on the mirror.

The following figure shows the law of reflection determining how this incident ray would reflect from the mirror.

As we recall from the law of reflection, the incident angle equals the reflection angle. Each of these angles is between a light ray and the line normal to the surface of the mirror.

Let us look at an example involving the reflection of a light ray from a concave mirror.

Example 1: Identifying the Path of a Ray Reflected from a Concave Mirror

The following figure shows three light rays incident on a concave mirror. Which is a correctly drawn reflected ray, ray A or ray B?

Answer

The correct path of the reflected ray is either that of ray A or of ray B shown in the question.

It would be easy to mistake how to draw the path for the correct ray by incorrectly comparing a concave mirror to a plane mirror.

Reflection of the incident ray shown in red from a plane mirror would be as shown in the following figure.

This would correspond to ray B in the question.

However, the incident ray is reflecting from a concave mirror, not a plane mirror.

This means that we can eliminate ray B. This only leaves ray A, which must be correct.

Let us consider why ray A is correct.

The following figure shows that the ray shown in red is symmetrical about the optical axis of the mirror to the ray shown in blue.

The rays shown in red and in blue are parallel to the optical axis, at equal distances from it. The angle made where the red incident ray crosses the blue reflected ray is equal to the angle made where the blue incident ray crosses the red reflected ray; ray A.

When drawing ray diagrams with mirrors, usually we may consider parallel incident rays.

The following figure shows three parallel incident rays and how each ray reflects from a concave mirror.

We see that all the reflected rays pass through a point that is on the optical axis of the mirror.

There are two special points on the optical axis of any curved mirror. These points are called

• the center of curvature of the mirror,
• the focal point of the mirror.

The center of curvature of a mirror is a point that is at the same distance from the surface of the mirror in every direction.

The focal point of a curved mirror is a point at which the reflected rays of parallel incident rays all cross each other’s paths.

Let us look at an example involving the reflection of parallel incident light rays from a concave mirror.

Example 2: Identifying a Point along the Optical Axis of a Concave Mirror

The following figure shows three parallel light rays incident on a concave mirror. Which of the following is the term for point P, which is shown by the black dot?

1. Focal point of the mirror
2. Center of curvature of the mirror

Answer

The center of curvature of a mirror is a point that is at the same distance from the surface of the mirror in every direction.

In the following figure, we can compare the lengths of the two dashed lines from P to different parts of the surface of the mirror.

It is clear that these two dashed lines have unequal lengths. This tells us that P is not the center of curvature of the mirror.

The focal point of a curved mirror is a point at which the reflected rays of parallel incident rays all cross each other’s paths.

We see that all the incident rays are parallel. We see also that all the reflected ray paths cross at P.

Therefore, P is the focal point of the mirror.

Suppose that an object is placed in front of a concave mirror. We can consider the light from one point on the object.

The following figure shows light rays from a point on an object that are incident on a concave mirror.

The light rays from the point travel in different directions.

The following figure shows how these two rays reflect from the mirror.

Two special points are marked. Point A is the point from which light rays start.

We can see that at the other special point, point B, the paths of the two reflected rays cross each other.

We see that point B is at the opposite end of the object to point A.

We have only shown two directions in which light rays from point A could travel.

It is very important to understand though that light rays from point A that travel in any direction will meet at point B.

It is important to understand that this is only true for light rays that reflect from the mirror. A light ray from point A that travels to the left will not reflect from the mirror and so will not arrive at point B.

This means that all light rays from point A that are reflected by the mirror will meet at point B.

This means that at point B, a real image is formed of the part of the object at point A.

This tells us that if a screen was placed at point B, we would be able to see on the screen whatever was at point A.

We can apply this to every point on an object.

Light rays from each point on the object will be reflected to another point, forming an image of that point.

Each different point on an object produces an image at a different point. Light rays from points on an object that are next to each other are reflected to points on an image that are next to each other.

This means that an image of an entire object is formed.

We can see here an image formed by a concave mirror.

We cannot actually see the object that produced the image.

We can reasonably guess, however, that the incident light rays on the mirror came from the sky, some trees, and the ground; we see these things in the image.

It is important to notice that the image is upside down. The sky is at the bottom of the mirror and the ground is at the top.

This should not surprise us when we recall how light rays are reflected from a point by a concave mirror, shown in the following figure.

The image is of an image of point A, which is at the top of the object, is at point B, which is at the bottom of the object.

We see then that a concave mirror can produce an inverted image of an object.

The image produced by a concave mirror changes depending on how far in front of the mirror an object is.

We have seen examples so far of an object that is at a distance from the back of the mirror equal to the distance from the back of the mirror to the center of curvature of the mirror. This is shown in the following figure.

We have seen that for an object at this distance from the back of the mirror, the image is inverted.

We can see that a point at the top of the object appears in the image at the bottom of the object.

The following figure shows that this means that the image must be the same size as the object.

If an object is moved toward or away from the back of a concave mirror, this changes the size of the image.

The formation of images for objects behind and in front of the center of curvature of a concave mirror is shown in the following figure.

We can see that the images produced are still inverted.

We can see though that for both cases, the vertical distance from the center of curvature to point A is not equal to the vertical distance from the center of curvature to point B.

This tells us that in both these cases, the size of the image is not equal to the size of the object.

Let us now look at an example involving the image of an object formed by reflection from a concave mirror.

Example 3: Comparing the Size of an Image Formed by a Concave Mirror to an Object

The following figure shows two light rays from the same point on an object that are incident on a concave mirror. The object is between the center of curvature of the mirror and the focal point of the mirror. A real image is produced. Which of the following is true?

1. The image is larger than the object.
2. The object is larger than the image.
3. The image and the object are the same size.

Answer

The image formed by a concave mirror for an object located between the center of curvature and focal point of the mirror is an inverted image.

The top of the image is at the point where the paths of light rays reflected from the top of the object cross each other.

We can see two things about this point.

• The point is further from the back of the mirror than the distance from the surface of the mirror to its center of curvature.
• The vertical distance from the center of curvature to the top of the image is greater than the vertical distance from the center of curvature to the top of the object.

The image would appear with the size and position shown in the following figure.

We see that the image is larger than the object.

It would not be possible in practice to see all of this image, as the object would block some of the light rays that would be needed to form the image.

We can see that there is a pattern relating the distance of an object from the back of a lens and the size of the image produced.

• When an object is located between the focal point and center of curvature of a concave mirror, the image is larger than the object.
• When an object is located at the center of curvature of a concave mirror, the image is the same size as the object.
• When an object is located further from the back of a concave mirror than the distance from the center of curvature to the surface of the mirror, the image is smaller than the object.

If an object is located closer to the back of a concave mirror than the distance from the focal point to the back of the mirror, an interesting thing happens. This is shown in the following figure.

We see that none of the paths of the reflected rays cross each other. This means that no real image is formed.

However, if the paths of the reflected rays are traced back, they meet at a point.

The top of the point that the reflected ray paths meet at is the top of a virtual image.

This is shown in the following figure.

We can see that the virtual image is the same way up as the object.

We can see that the virtual image is larger than the object.

A virtual image cannot form on a screen but can be seen by the human eye.

The fact that a concave mirror can make inverted real images and upright virtual images means that a concave mirror can show some objects inverted and others upright at the same time, as shown below.

Let us now summarize what has been learned in this explainer.