Convex Mirror — Properties, Ray Rules, Formula, Uses, Practice Problems & FAQs
A concise and original explanation of convex (diverging) mirrors: how they form images, the standard ray rules, mirror equation and magnification, practical applications, four fresh practice problems (with full solutions), and common questions answered.
Spherical mirrors — quick recap
A spherical mirror is a portion of a reflective spherical surface. Two standard types exist: concave (inner surface reflective — converging) and convex (outer surface reflective — diverging). In this article we focus on convex mirrors, which give a wide field of view and always form virtual images.
What is a convex mirror?
Imagine a hollow glass sphere coated with silver on the inside; when you look at the outside surface, that is a convex mirror. Because its reflecting surface bulges toward the incoming light, reflected rays spread (diverge) and the mirror is often called a diverging mirror.
Key characteristics of convex mirrors
- Reflected rays diverge after reflection — they never meet in front of the mirror.
- The image produced is always virtual, upright (erect), and diminished (smaller than the object).
- As the object moves closer or farther, the virtual image moves along the line between the pole and the focus but remains behind the mirror.
Important terms
- Pole (P): the geometrical centre of the mirror surface.
- Centre of curvature (C): centre of the sphere of which the mirror is a part.
- Principal axis: the straight line through P and C.
- Focus (F): the point from which the reflected rays appear to diverge (for convex mirrors the focus is behind the mirror).
- Radius of curvature (R): the radius of the parent sphere (distance PC).
- Focal length (f): distance PF (for our sign convention we will take f positive for a convex mirror and measure u, v with signs explained below).
Ray-tracing rules (useful for drawing a ray diagram)
To locate the image of an object using a convex mirror, draw at least two of these standard rays from a point on the object:
- Parallel→apparent focus: A ray parallel to the principal axis reflects as if it originated from the focus behind the mirror.
- Toward focus→parallel: A ray aimed toward the focus (if extended behind the mirror) reflects and emerges parallel to the principal axis.
- Towards centre→back on itself: A ray directed toward the centre of curvature (behind the mirror) reflects back along the same path.
- Incidence at pole: A ray striking the pole reflects with the angle of incidence equal to the angle of reflection (like a plane mirror) and can be used as a check.
Convex mirror — ray diagram
sample ray diagram showing two reflected rays extended behind the mirror to locate the virtual image.
Image formation (qualitative)
Because reflected rays from a convex mirror diverge, they never form a real image in front of the mirror. If you extend the diverging rays backward (behind the mirror), they meet at a point that serves as the virtual image location. That image is always upright and smaller than the object, which is why convex mirrors are ideal where a wide field of view is desired.
Mirror formula & magnification
The mirror equation relates object distance \(u\), image distance \(v\), and focal length \(f\):
Magnification \(m\) is the ratio of image height \(h_i\) to object height \(h_o\). In terms of distances:
Sign convention used here (consistent with the worked examples below): take the direction of incident light as positive. For a real object placed in front of the mirror we use \(u<0\); a virtual image located behind the mirror corresponds to \(v>0\); for convex mirrors we treat \(f>0\).
Common uses of convex mirrors
- Vehicle side-mirrors (wide field of view and reduced blind-spots).
- Safety mirrors at blind corners on roads and in stores to monitor aisles.
- Entrance/security mirrors and decorative applications where a panoramic view is useful.
Practice problems (new numbers) — worked solutions
Given (using our sign convention): \(u=-30.0\ \text{cm}\) and \(v=+12.0\ \text{cm}\).
Use the mirror formula:
Compute (common denominator 60): \(-2/60 + 5/60 = 3/60 = 1/20\). So
A positive focal length with the image behind the mirror indicates the mirror is convex.
Answer: Convex mirror, focal length \(f=20.0\ \text{cm}.\)
Given \(f=15.0\ \text{cm}\) and \(u=-28.0\ \text{cm}\). Solve for \(v\):
Compute: common denominator 420 → \(28/420 + 15/420 = 43/420\). Thus
Since \(v>0\), the image is virtual and located about \(9.77\ \text{cm}\) behind the mirror.
Answer: \(v \approx 9.8\ \text{cm}\) (virtual, behind mirror).
Magnification \(m = h_i/h_o = 4.0/12.0 = 0.333\). Because the image height is positive (virtual & upright), the image is erect and diminished.
Answer: \(m = 0.333\) (image is upright / erect).
Using distances \(v=+20.0\ \text{cm}\) and \(u=-60.0\ \text{cm}\), magnification:
Positive magnification indicates an upright virtual image, with size one third of the object.
Answer: \(m = 0.333\) (upright, diminished).
FAQs — quick answers
Q1. Why are convex mirrors used as vehicle side mirrors?
A: They provide a wider field of view and create a smaller, upright virtual image so the driver can see more area behind and to the side of the vehicle.
Q2. Does a convex mirror ever form a real image?
A: No — a convex mirror’s reflected rays always diverge, so any image is virtual (located by extending rays behind the mirror).
Q3. Why is magnification positive for convex mirrors?
A: The image produced by a convex mirror is upright relative to the object; with our sign convention this corresponds to a positive magnification \(m=h_i/h_o>0\).
Q4. How does magnification change as an object moves farther away?
A: For a convex mirror the virtual image gets closer to the focus behind the mirror and becomes progressively smaller; magnification decreases (image shrinks) as object distance increases.
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