E. Physics: Light and Geometric Optics · Light and Geometric OpticsRay Diagrams for Convex Mirrors and Their Uses
E2.3 E3.3 E1.2 A1.11About 65 minutes
Learning goals- I can draw ray diagrams for convex mirrors and locate the virtual image.
- I can describe the image formed by a convex mirror using SALT.
- I can explain why convex mirrors have a wide field of view.
- I can assess the benefits and limitations of convex mirrors in everyday technology.
Words to know
diverging mirror
Another name for a convex mirror; it spreads parallel rays apart.
virtual focal point
For a convex mirror, the point behind the mirror from which reflected parallel rays appear to come.
field of view
The region that can be seen in a mirror or through an optical device.
blind spot
An area around a vehicle that the driver cannot see in the mirrors.
shoulder check
Turning your head to look into a blind spot before changing lanes.
Objects in mirror are closer than they appear
The mirror on the passenger side of many cars has a warning printed on it: "Objects in mirror are closer than they appear." That mirror is convex. Convex mirrors show a wider view than flat mirrors, but everything looks smaller, and smaller things look farther away. In this lesson you will see exactly why, with ray diagrams.
Rules for convex-mirror ray diagrams
For a convex mirror, F and C are behind the mirror. The rules are the concave rules, adjusted so the rays head towards or appear to come from points behind the mirror:
- A ray parallel to the principal axis reflects as if it came from F. (Draw a dashed line from F to the point where it hits the mirror.)
- A ray heading towards F reflects parallel to the principal axis.
- A ray heading towards C reflects straight back on itself (it travels along a normal).
The reflected rays always spread apart. Extend them backwards with dashed lines: where the extensions meet, behind the mirror, is the virtual image.
Worked exampleExample 1: Let's draw a convex-mirror diagram
A convex mirror has a focal length of magnitude 3.0 cm. A 3.0 cm tall object stands 6.0 cm in front of it. Draw a scale diagram and describe the image.
- Draw the principal axis and the mirror. Mark F 3.0 cm behind the mirror and C 6.0 cm behind it.
- Draw the object 6.0 cm in front of the mirror, 3.0 cm tall.
- Ray 1: from the tip, parallel to the axis to the mirror. It reflects as if from F: line up a ruler with F and the point where the ray hits, and draw the reflected ray outward. Draw a dashed line back to F.
- Ray 2: from the tip towards F. Stop at the mirror; it reflects parallel to the axis. Extend the reflected ray backwards (dashed) behind the mirror.
- Ray 3: from the tip towards C. It reflects back along itself. Its dashed extension continues to C.
- The dashed lines meet 2.0 cm behind the mirror, 1.0 cm above the axis. Draw a dashed upright arrow there.
- SALT: smaller (1.0 cm vs 3.0 cm), upright, 2.0 cm behind the mirror (between V and F), virtual.
The image is smaller, upright, 2.0 cm behind the mirror, and virtual.
The convex mirror always does the same thing
Try a few different object positions and you will find that a convex mirror always forms an image that is:
- S: smaller than the object,
- A: upright,
- L: behind the mirror, between V and F (closer to F for distant objects, closer to V for near objects),
- T: virtual.
The farther away the object, the smaller its image and the closer the image is to F.
Why convex mirrors are used: a wider field of view
The field of view is the region you can see in a mirror. For a plane mirror, the edges of the field of view are set by the law of reflection at the mirror's edges. A convex mirror's surface tilts outward towards its edges, so rays reflected from the edges spread out much more. A convex mirror of the same size, seen from the same place, shows you a much wider area.
Convex mirrors in society
Convex mirrors make the world a little safer:
- Vehicle mirrors. Passenger-side mirrors on many cars and the large mirrors on school buses and trucks show more of the road and the blind spots beside the vehicle.
- Blind corners. Convex mirrors at the exits of parking garages, at sharp corners in hallways and at blind driveways let people see traffic coming around the corner.
- Stores. Security mirrors in the corners of stores let one employee watch many aisles.
The trade-off: because the image is smaller, our brain judges objects to be farther away than they really are. That is the reason for the warning on car mirrors, and it is why drivers are taught to shoulder check before changing lanes. A good technology assessment always weighs both the benefits and the limitations.
| Plane mirror | Concave mirror | Convex mirror |
|---|
| Shape | flat | curves inward | bulges outward |
| Parallel rays | stay parallel | converge (real F) | diverge (virtual F) |
| Possible images | same size, upright, virtual | real & inverted (object beyond F) or virtual, upright & larger (object inside F) | always smaller, upright, virtual |
| Field of view | normal | narrow | wide |
| Typical uses | bathroom mirror, periscope | makeup mirror, headlight, telescope | car and bus mirrors, store security mirror |
Just for funA convex mirror walks into a corner store. The owner looks up and says, "Finally, someone around here who sees the big picture."
Hands-on activitySpoon versus flat mirror: field of view
You need: a shiny metal spoon, a flat mirror about the same size as the bowl of the spoon (for example a compact mirror), a ruler.
- Hold the flat mirror at arm's length and count how many objects in the room behind you that you can see in it (or how much of a wall).
- Now hold the back of the spoon at the same distance and count again.
- Hold the back of the spoon at 10 cm, 20 cm and 40 cm from your eye. Each time, describe your image with SALT. Does the image ever become inverted or larger?
- Write a short evaluation: if you were designing a mirror for the exit of an underground parking garage, would you choose plane or convex? Give one benefit and one limitation.
What to look for: the spoon's back shows far more of the room; your image is always smaller, upright and virtual, getting smaller as you move the spoon away. Convex is the better choice for the garage (wide view), but drivers must remember that cars look farther away than they are.
Your turn
1Where are F and C for a convex mirror?
Behind the mirror (on the opposite side from the object).
2Describe the image formed by a convex mirror using SALT.
Smaller, upright, behind the mirror between V and F, virtual.
3For a convex mirror, how does a ray that is heading towards F reflect?
It reflects parallel to the principal axis.
4Why are reflected rays from a convex mirror extended with dashed lines?
The reflected rays diverge and never meet in front of the mirror. Their backward extensions (where no light actually travels) meet behind the mirror to locate the virtual image.
5Explain why "objects in mirror are closer than they appear" in a convex car mirror.
The convex mirror makes smaller images. Our brain interprets a smaller image of a car as a car that is farther away, so cars are closer than they look.
6Give two places where convex mirrors improve safety.
For example: passenger-side and bus mirrors, parking-garage exits, blind corners and driveways, store aisles.
More practice: warm-up
1Is a convex mirror converging or diverging?
2Can a convex mirror form a real image?
No (not of a real object): its images are always virtual.
3Is the image in a convex mirror larger or smaller than the object?
4Does a convex mirror give a wider or narrower field of view than a plane mirror of the same size?
5Which part of a spoon acts as a convex mirror?
More practice: on your own
1Explain, using normals, why a convex mirror gives a wide field of view.
The normals at a convex mirror's edges tilt outward (they point away from C, which is behind the mirror). Rays reflected near the edges are sent farther outward, so the mirror collects light from a much wider region.
2An object moves from 2 m to 20 m in front of a convex mirror. What happens to its image?
The image stays upright and virtual but becomes smaller and moves closer to F (farther behind V).
3A student draws a convex-mirror ray diagram and finds an inverted image in front of the mirror. What did the student probably do wrong?
They probably treated the mirror as concave: drawing rays through F and C as if those points were in front of the mirror. For a convex mirror, F and C are behind it and rays only head towards or appear to come from them.
4Evaluate convex mirrors as store security devices: give one benefit and one limitation.
Benefit: one employee can see many aisles at once because of the wide field of view. Limitation: images are small and distorted, so details (such as a face) are hard to see; cameras may do the job better.
5A convex mirror and a plane mirror are the same size. Which shows your face larger, and why?
The plane mirror: its image is the same size as you; the convex mirror's image is smaller.
Challenge
1Use the scale diagram method with f = −2.0 cm and an object 4.0 cm tall at 2.0 cm from a convex mirror. Use the special rays (a scale diagram or careful reasoning) to predict the image location and size.
The rays' extensions meet 1.0 cm behind the mirror, halfway to F, and the image is 2.0 cm tall, upright and virtual. (Check: ray 1's extension runs from the hit point 4.0 cm up at V down to F at 2.0 cm behind; at 1.0 cm behind it is 2.0 cm high. Ray 2 aims at F, 4.0 cm away horizontally from the tip, so it hits the mirror at 2.0 cm high and reflects parallel; its extension is at 2.0 cm high. They meet at 1.0 cm behind, 2.0 cm high.)
2A driver sees a car in a convex side mirror. The image is 1⁄10 the size it would be in a plane mirror. Explain why judging distance with this mirror is risky, and suggest a design solution used on some vehicles.
The brain uses image size to judge distance, so a car whose image is 10 times smaller seems much farther away than it is. Solutions include printing a warning, using a plane mirror for the driver's side, using mirrors that are flat in the centre and convex at the edge (aspheric mirrors), or adding blind-spot sensors and cameras.
3Can a convex mirror ever make a real image? Explain using the rules for rays.
Not for a real object in front of it. Every ray from the object is reflected so that it spreads away from the axis (as if from F, or parallel, or back towards the object), so the reflected rays never cross in front of the mirror. Only their backward extensions meet, giving a virtual image.
Quiz
1. The image formed by a convex mirror is always:
2. For a convex mirror, a ray parallel to the principal axis reflects:
3. Why are convex mirrors used at parking-garage exits?
4. For a convex mirror, the image is located:
5. A ray heading towards C of a convex mirror:
6. "Objects in mirror are closer than they appear" is printed on convex mirrors because:
Big idea: Convex mirrors always form smaller, upright, virtual images and give a wide field of view (E2.3, E3.3). Expectation E1.2 asks students to analyse how such devices benefit society, including their limitations.
Common difficulties:
- Drawing rays "through" F and C. For convex mirrors these points are behind the mirror, so rays only head towards them or appear to come from them.
- Forgetting dashed extensions. The image is found only where the dashed lines meet.
- Sign of f. In the next lesson, convex mirrors get a negative focal length. Point out now that F is on the "virtual" side.
How to help: On your next drive, look at the passenger-side mirror together and compare it with the driver's-side or rear-view mirror. Ask which shows more, and which makes cars look farther away.