Ray Diagrams for Spherical Mirrors

Hello! Welcome to the third lesson in our exploration of ray optics.

In our previous session, we established the crucial geometric relationship $f = R /2$ , linking a mirror's focal length to its radius of curvature. This told us precisely where to locate the principal focus (F) relative to the center of curvature (C) and the pole (P).

Today, we will use this foundation to build a powerful visual tool: ray tracing. Your learning outcome for this lesson is to be able to construct ray diagrams for image formation by concave and convex mirrors. By mastering this graphical method, you'll be able to predict the location, size, and nature (real/virtual, upright/inverted) of an image for any object position. This geometric approach will provide an intuitive understanding that will perfectly complement the algebraic formulas we'll derive in our next lesson.

1. The Rules of the Game: Principal Rays

An object reflects countless rays of light in all directions. To find where its image is formed, we only need to trace the paths of a few special, predictable rays called principal rays. The point where these reflected rays intersect (or appear to diverge from) locates the image.

There are four principal rays that are particularly easy to draw because their behavior upon reflection is governed by simple rules. Any two are sufficient to locate an image point, but using a third can be a good way to verify your drawing.

To begin, study the following text, which defines these four principal rays. Then, use the diagram below it as a quick visual reference.

Spherical Mirrors

The OpenStax 'University Physics' textbook provides a clear, formal definition of the four principal rays used in ray tracing. This section will give you the foundational rules for all the diagrams we will construct today.

Please read the section titled 'Using Ray Tracing to Locate Images'. Focus on the descriptions of 'Principal ray 1' through 'Principal ray 4' and the final summary box titled 'Ray-Tracing Rules'.

Here is a summary diagram illustrating these four essential rules for both concave and convex mirrors. Keep this handy as we proceed.

Ray Tracing Rules for Concave and Convex Mirrors — This diagram shows the four principal rays: 1. A ray parallel to the axis (P-F ray), 2. A ray through the focus (F-P ray), 3. A ray through the center of curvature (C-C ray), and 4. A ray striking the vertex (V ray). Notice how for a convex mirror, the rays on the right side are extended as virtual lines (dashed) to locate the virtual focus and center of curvature.

2. Image Formation by a Concave Mirror

Now, let's apply these rules. A concave mirror is fascinating because the characteristics of the image it forms change dramatically depending on where the object is placed. We will investigate five key cases.

For each case, we'll use the precise and clear animations from Khan Academy to see the diagram being constructed.

Representation of images formed by spherical mirrors using ray diagrams | Physics | Khan Academy

This Khan Academy video provides excellent, quick demonstrations of how to draw the ray diagrams for each object position. We will watch it in segments as we cover each case.

You don't need to watch the whole video at once. I will point you to the specific timestamps for each case below.

Case 1: Object is placed beyond the Center of Curvature (u > 2f)

Watch: Khan Academy video from 00:00 to 01:28.
Construction: A ray parallel to the axis reflects through F. A ray passing through F reflects parallel to the axis.
Image Properties:
- Location: Between the focus (F) and the center of curvature (C).
- Nature: Real (the reflected rays actually intersect) and Inverted.
- Size: Diminished (smaller than the object).

Case 2: Object is placed at the Center of Curvature (u = 2f)

Watch: Khan Academy video from 01:44 to 02:20.
Construction: A ray parallel to the axis reflects through F. A ray passing through F reflects parallel.
Image Properties:
- Location: At the center of curvature (C).
- Nature: Real and Inverted.
- Size: Same size as the object.

Case 3: Object is placed between C and F (f < u < 2f)

Watch: Khan Academy video from 01:28 to 01:44.
Construction: Again, we use the ray parallel to the axis reflecting through F, and the ray through F reflecting parallel.
Image Properties:
- Location: Beyond the center of curvature (C).
- Nature: Real and Inverted.
- Size: Magnified (larger than the object).

Case 4: Object is placed at the Principal Focus (u = f)

Watch: Khan Academy video from 03:13 to 03:50.
Construction: A ray parallel to the axis reflects through F. A ray striking the vertex reflects symmetrically.
Image Properties: The reflected rays are parallel. They never intersect, so no image is formed (or we say the image is formed at infinity). This principle is used in searchlights or car headlights to produce a powerful, parallel beam of light.

Case 5: Object is placed between F and the Pole (u < f)

Watch: Khan Academy video from 02:20 to 03:13.
Construction: A ray parallel to the axis reflects through F. A ray striking the vertex reflects symmetrically. The reflected rays diverge. To find the image, we trace them backward (as dashed lines) behind the mirror.
Image Properties:
- Location: Behind the mirror.
- Nature: Virtual (the rays only appear to originate from this point) and Upright.
- Size: Magnified. This is the principle behind a shaving mirror or a makeup mirror.

Here is a summary table of our findings for the concave mirror:

Object Position	Image Position	Image Nature	Image Size
At Infinity	At F	Real, Inverted	Point-sized
Beyond C	Between F & C	Real, Inverted	Diminished
At C	At C	Real, Inverted	Same size
Between C & F	Beyond C	Real, Inverted	Magnified
At F	At Infinity	—	—
Between F & P	Behind mirror	Virtual, Upright	Magnified

3. Image Formation by a Convex Mirror

Image formation by a convex mirror is much simpler. Because the focus F and center of curvature C are located behind the mirror's reflecting surface, the outcome is the same regardless of where the object is placed.

Watch: Khan Academy video from 03:50 to 04:53.
Construction:
1. A ray parallel to the principal axis reflects as if it came from the virtual focus F.
2. A ray directed towards the center of curvature C reflects back on itself.
3. The reflected rays always diverge. We trace them backward (as dashed lines) to find their apparent point of origin.
Image Properties (for any object position):
- Location: Between the pole (P) and the virtual focus (F).
- Nature: Virtual and Upright.
- Size: Diminished.

This consistent behavior is why convex mirrors are used for security and as side-view mirrors on vehicles—they always provide a wide field of view with an upright (though smaller) image.

Test your understanding!

Consider a concave mirror. An object is placed at a distance from the pole equal to three times the focal length (u = 3f).

In which region (e.g., "between C and F") is this object located?
Using a quick sketch with at least two principal rays, determine the properties of the image (Location, Nature, Size).

Show answer

We know that the center of curvature C is at a distance $R = 2 f$ from the pole. Since the object is at $u = 3 f$ , it is located beyond the center of curvature (C).
This corresponds to Case 1 for a concave mirror.
- A ray from the top of the object parallel to the principal axis will reflect through the focus F.
- A ray from the top of the object passing through the center of curvature C will reflect back on itself.
- The intersection of these two reflected rays will occur between F and C.
- The image formed will be Real, Inverted, and Diminished.

Conclusion

In this lesson, you've learned the essential skill of constructing ray diagrams to analyze image formation by spherical mirrors.

Key Takeaways:
- Image location and properties can be determined by tracing principal rays. The four key rays are the P-F, F-P, C-C, and V-rays.
- Concave mirrors are versatile, capable of forming both real, inverted images and virtual, upright images depending on the object's position relative to F and C.
- Convex mirrors are consistent: they always produce a virtual, upright, and diminished image, regardless of the object's position.

Preview of the Next Lesson:
Ray diagrams provide an excellent qualitative understanding, but they are limited by the accuracy of our drawings. For precise, quantitative answers, we need algebra. In the next lesson, we will derive two of the most important equations in geometric optics: the Mirror Formula ( $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ ) and the Magnification Formula ( $m = - \frac{v}{u}$ ). These will allow you to calculate the exact position and size of an image, turning your graphical sketches into exact numerical solutions.

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