Light – Reflection and Refraction · Lesson 4 of 15
Representation of Images Formed by Spherical Mirrors Using Ray Diagrams
“Two rays do all the detective work while the others enjoy the day off.”
• Explain why two suitable rays are enough to locate an image. • Apply the four standard ray rules for spherical mirrors. • Use solid and extended rays correctly. • Construct and interpret mirror ray diagrams. • Recognise common geometrical errors in ray diagrams.
Every point on an illuminated object sends out many rays, but drawing all of them would hide the geometry we need. A ray diagram chooses two rays whose paths after reflection are easy to predict. Their intersection locates the image of the chosen object point.
Why Two Rays Are Sufficient
If two reflected rays from the tip of an object actually intersect, the tip of a real image is at that intersection. If they diverge, extend them backward with dashed lines; where those extensions meet is the tip of a virtual image. A third correct ray would pass through the same image point and can be used as a check.
Standard Ray Rules
A ray parallel to the principal axis passes through principal focus after reflection from a concave mirror or appears to come from principal focus after reflection from a convex mirror.
A ray passing through principal focus of a concave mirror, or directed toward principal focus of a convex mirror, emerges parallel to the principal axis.
A ray passing through centre of curvature of a concave mirror or directed toward centre of curvature of a convex mirror, retraces its path because it strikes the mirror normally.
A ray striking the pole reflects so that its angles with the principal axis are equal, in accordance with the laws of reflection.
The ray through C returns along itself because a radius to a spherical surface is perpendicular to the tangent at the point of incidence. Its incidence angle is therefore zero. The pole rule is useful when F or C is inconvenient to use.
A Reliable Construction Method
- Draw a horizontal principal axis and sketch the mirror with its reflecting side clear.
- Place P on the mirror and mark F and C at correct relative distances, with PF half of PC.
- Draw the object as an upright arrow with its base on the principal axis.
- From the object tip, draw any two standard incident rays and apply the matching reflection rules.
- Use solid lines for actual rays. Use dashed backward extensions only when the reflected rays diverge.
- Mark the image from the axis to the intersection and state its position, orientation, size and nature.
Problem
An object is beyond C in front of a concave mirror. Predict the image using a parallel ray and a ray through C.
- 1.The parallel ray reflects through F.
- 2.The ray through C retraces its path.
- 3.The two reflected rays meet between F and C.
- 4.Their actual intersection gives a real image; the image arrow points below the axis, so it is inverted and diminished.
Problem
An object is between P and F of a concave mirror. Explain how its image is located.
- 1.Draw a ray parallel to the axis; after reflection it travels through F.
- 2.Draw a ray toward C; after reflection it returns along its path.
- 3.The reflected rays diverge in front of the mirror, so extend them backward behind the mirror with dashed lines.
- 4.Their extensions meet behind the mirror, giving a virtual, erect and enlarged image.
Problem
For a convex mirror, show why a finite object always gives a virtual diminished image.
- 1.A parallel incident ray reflects as though it came from F behind the mirror.
- 2.A ray directed toward C retraces its path after reflection.
- 3.The reflected rays diverge; only their backward extensions meet behind the mirror.
- 4.The intersection lies between P and F and is closer to the axis than the object tip, so the image is virtual, erect and diminished.
Do not measure P, F and C from the mirror's edge; use the pole. Do not draw backward extensions as solid rays. Do not bend a ray before it reaches the mirror, and do not measure reflection angles from the mirror surface.
Quiz
What locates the image point in a real-image ray diagram?
What happens to a concave-mirror ray passing through C?
How should backward extensions be drawn?
A ray parallel to the axis of a convex mirror reflects as if it came from where?
Why is a third correctly drawn ray useful?
Practice Problems
- State the reflected path of a ray passing through F of a concave mirror. Solution: It emerges parallel to the principal axis.
- Explain why a ray through C retraces its path. Solution: It strikes along the radius, which is normal to the spherical surface, so the incidence and reflection angles are both zero.
- Construct verbally the image for an object at C. Solution: A parallel ray reflects through F; a ray through C retraces. They meet at C below the axis, producing a real, inverted, same-sized image.
- How can a drawing reveal that an image is virtual? Solution: The actual reflected rays diverge, while dashed backward extensions meet behind the mirror.
- A student places F farther from P than C. Diagnose the error. Solution: For a small-aperture spherical mirror, F must lie midway between P and C because R = 2f.
Key Takeaways
• Two well-chosen rays are sufficient to locate an image point. • Standard rays use the predictable roles of F, C and P. • Actual ray intersections give real images. • Backward extensions of diverging rays locate virtual images. • Solid lines represent actual light paths; dashed lines represent extensions. • Correct placement of P, F and C is essential for an accurate diagram.