Ray Tracing with the Law of Reflection

1. The Fundamental Law of Reflection

All of geometric optics is built upon a few simple principles. The most basic of these is the law of reflection, which describes how light bounces off a surface.

The law states:

1. The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
2. The angle of incidence is equal to the angle of reflection.

Let's define these terms visually:

It is critical to remember that these angles are always measured relative to the normal, not the mirror surface.

To see this principle in action and how it's used to solve simple geometry problems, let's watch a short video.

As the video showed, even in a more complex setup with two mirrors, the solution comes from applying the same simple rule (${\theta }_i={\theta }_r$) at each reflection point and using your knowledge of geometry.

2. Ray Tracing for Spherical Mirrors

While the law of reflection is simple, applying it to every single point on a curved mirror would be tedious. Fortunately, for spherical mirrors (mirrors that are a section of a sphere), we can identify a few "principal rays" whose paths are very easy to predict. By tracing just two of these rays from an object, we can determine exactly where its image will be formed.

Before we learn the rules, let's define the key parts of a spherical mirror system:

• Pole (P): The geometric center of the mirror's surface.
• Center of Curvature (C): The center of the sphere from which the mirror was cut. Any line from C to the mirror is a radius and is normal to the mirror surface at that point.
• Principal Axis: The straight line passing through the Pole (P) and the Center of Curvature (C).
• Principal Focus (F): The point on the principal axis where rays initially parallel to the axis converge (concave mirror) or appear to diverge from (convex mirror) after reflection. It is located halfway between P and C, so the focal length $f$ is half the radius of curvature $R$ ($f=R/2$).

To get a solid grasp of these rules, we'll use a combination of video and a concise summary document. The video will help you visualize the ray paths, and the document will serve as a quick reference.

Rules for Concave Mirrors (Converging Mirrors)

Let's start with concave mirrors.

Now, let's consolidate those rules with a clear, written summary.

Here is a summary of the four essential rules for tracing rays in a concave mirror:

1. A ray parallel to the principal axis is reflected to pass through the focus (F).
2. A ray passing through the focus (F) is reflected parallel to the principal axis. (This is the reverse of Rule 1).
3. A ray passing through the center of curvature (C) reflects back along the same path. This is because it strikes the mirror along a radius, which is normal to the surface, making the angle of incidence 0°.
4. A ray striking the pole (P) is reflected symmetrically, with the principal axis acting as the normal.

Rules for Convex Mirrors (Diverging Mirrors)

The rules for convex mirrors are very similar, but because the focus and center of curvature are behind the mirror, the rays only appear to come from or go to these points.

Let's watch the next segment of the video.

To reinforce this, please refer back to the Rules for obtaining images formed by spherical mirrors document (LINK) and review the diagrams for the convex mirror rules. They are presented alongside the concave mirror rules.

Here is a summary of the four rules for a convex mirror:

1. A ray parallel to the principal axis is reflected so that it appears to have come from the focus (F).
2. A ray directed towards the focus (F) is reflected parallel to the principal axis.
3. A ray directed towards the center of curvature (C) reflects back along the same path.
4. A ray striking the pole (P) is reflected symmetrically.

Mastering these eight rules (four for each mirror type) is the key to all ray diagrams.

3. A Glimpse into Image Formation

So, what is the point of tracing these rays? By taking any two of these principal rays from a single point on an object (e.g., the tip of an arrow), their intersection point after reflection will show you the location of that point on the image.

The video Spherical Mirrors goes on to show how this is done for various object positions (starting at 05:04 for concave and 16:10 for convex). You don't need to study these in detail right now, as that is the focus of our next lesson. However, watching one example will show you how the rules you've just learned are put into practice. The first case in the video (object beyond C for a concave mirror, from 06:17 to 07:30) is a perfect example of using Rule 1 and Rule 2 to locate an image.

Conclusion

In this lesson, we laid the essential groundwork for our study of mirrors. You've learned the fundamental law of reflection and, more importantly, the four principal ray-tracing rules for both concave and convex mirrors. These rules are simply convenient applications of the law of reflection that make drawing diagrams manageable.

Key Takeaways:

• The law of reflection states that the angle of incidence equals the angle of reflection (${\theta }_i={\theta }_r$), with angles measured from the normal.
• For spherical mirrors, we use four principal rays to simplify ray tracing.
• Rule 1 (Parallel Ray): A ray parallel to the principal axis reflects through (or appears from) the focus.
• Rule 2 (Focal Ray): A ray through (or towards) the focus reflects parallel to the principal axis.
• Rule 3 (Central Ray): A ray through (or towards) the center of curvature reflects back on itself.
• Rule 4 (Polar Ray): A ray hitting the pole reflects symmetrically.

Preview of the Next Lesson:
Now that you have the tools, our next lesson, "Construct ray diagrams to locate images formed by plane and spherical mirrors," will be all about using them. We will systematically place an object at different positions in front of concave and convex mirrors and use the rules you learned today to draw complete ray diagrams and determine the properties of the resulting image (its location, size, orientation, and whether it is real or virtual).