6.6 Image Formation by Spherical Mirrors
Spherical mirrors are curved mirrors formed from a section of a sphere. When the inner surface reflects light, it is a concave mirror, whereas when the outer surface reflects light, it is a convex mirror. These mirrors manipulate light rays differently, giving rise to distinct fields of vision, image positions, and image characteristics that are widely applied in optical instruments and everyday safety devices.
Learning Objectives
- Identify concave and convex spherical mirrors based on their reflecting surfaces.
- Compare the fields of vision of plane mirrors, concave mirrors, and convex mirrors.
- Define and explain optical terms related to spherical mirrors: principal axis, centre of curvature (\(C\)), radius of curvature (\(r\)), focal point (\(F\)), focal length (\(f\)), object distance (\(u\)), and image distance (\(v\)).
- State the relationship between the radius of curvature and focal length: \(r = 2f\).
- Apply rules of reflection to construct ray diagrams for concave and convex mirrors.
- Determine the position and characteristics of images formed by concave mirrors and convex mirrors at various object distances.
- Justify the applications of concave mirrors and convex mirrors in daily life.
Comparison of Ray Tracing and Image Formation in Concave vs Convex Mirrors
An all-in-one reference showing the three reflection rules, the primary ray diagrams, and field-of-vision comparisons.
Concave Mirror — Converging Mirror
Parallel rays converge at the real focal point, \(F\), in front of the mirror. The radius of curvature is twice the focal length: \(r = 2f\).
Convex Mirror — Diverging Mirror
Parallel rays appear to diverge from the virtual focal point, \(F\), behind the mirror. The convex mirror provides a wide field of vision.
Short Explanation
Introduction to Spherical Mirrors
Spherical mirrors are part of a sphere that has been cut. A concave mirror reflects light from its inner surface, while a convex mirror reflects light from its outer surface. A steel spoon demonstrates this: the inner surface acts as a concave mirror and the outer surface acts as a convex mirror.
Fields of Vision
For identical mirror sizes, a plane mirror has a narrow field of vision, a concave mirror has a very narrow field of vision, and a convex mirror has a wide field of vision.
Optical Terms
The principal axis passes through the centre of curvature, \(C\), and the pole, \(P\). The pole, \(P\) is the geometric centre of the mirror surface. The centre of curvature, \(C\) is the centre of the sphere. The radius of curvature, \(r\) is the distance between \(P\) and \(C\).
Focal Point and Focal Length
The focal point, \(F\) is where parallel rays converge for a concave mirror, or appear to diverge from for a convex mirror. The focal length, \(f\) is the distance between \(F\) and \(P\). The relationship is \(r = 2f\).
| Rule | Concave Mirror | Convex Mirror |
|---|---|---|
| Rule 1 | A light ray passing through \(C\) is reflected back along its original path. | A light ray travelling towards \(C\) is reflected back along its original path. |
| Rule 2 | A light ray parallel to the principal axis is reflected to the focal point, \(F\). | A light ray parallel to the principal axis is reflected as if it originates from the focal point, \(F\). |
| Rule 3 | A light ray passing through \(F\) is reflected parallel to the principal axis. | A light ray travelling towards \(F\) is reflected parallel to the principal axis. |
| Object Position | Image Position | Image Characteristics |
|---|---|---|
| At infinity | \(v = f\) (in front of mirror) | Real, inverted, diminished |
| Beyond \(C\) (\(u > 2f\)) | Between \(F\) and \(C\) (\(f < v < 2f\), in front of mirror) | Real, inverted, diminished |
| At \(C\) (\(u = 2f\)) | At \(C\) (\(v = 2f\), in front of mirror) | Real, inverted, same size as object |
| Between \(F\) and \(C\) (\(f < u < 2f\)) | Beyond \(C\) (\(v > 2f\), in front of mirror) | Real, inverted, magnified |
| At \(F\) (\(u = f\)) | At infinity (behind mirror / reflected rays are parallel) | Virtual, upright, magnified |
| Between \(F\) and \(P\) (\(u < f\)) | Behind mirror (\(v > u\)) | Virtual, upright, magnified |
| Convex Mirror Object Position | Image Position | Image Characteristics |
|---|---|---|
| Beyond \(F\) (\(u > f\)) | Behind mirror (\(v < f\)) | Virtual, upright, diminished |
| Between \(F\) and \(P\) (\(u < f\)) | Behind mirror (\(v < f\)) | Virtual, upright, diminished |
Applications of Concave Mirrors
Cosmetic mirror: produces an upright and magnified image when placed close to the face (\(u < f\)). Dental mirror: forms an upright and magnified virtual image (\(u < f\)). Reflector in car headlight: parabolic concave mirror reflects light rays from a bulb placed at its focal point to maintain parallel beams and light intensity over long distances.
Applications of Convex Mirrors
Blind spot mirror: placed at sharp corners of roads to widen the driver's field of vision. Security mirror in buildings: installed in buildings or shopping centres for surveillance purposes due to its wide field of vision. Vehicle rear-view / side mirrors: provide a wide field of vision enabling drivers to see vehicles approaching from behind.
Try to Answer First
Answer in your mind, then press “Check Answer”.
What is the mathematical relationship between the radius of curvature, \(r\), and the focal length, \(f\), of a spherical mirror?
Check Answer
Why are convex mirrors chosen as blind spot mirrors on sharp road corners instead of plane mirrors?
Check Answer
A person looks into the concave surface of a spoon at a distance of 30 cm and sees an inverted image. What happens to the image characteristics if the spoon is brought very close to the eyes (\(u < f\))?
Check Answer
Common Mistakes
- Confusing the reflecting sides: forgetting that the reflecting surface of a concave mirror curves inwards (like an entrance to a cave), while a convex mirror bulges outwards.
- Drawing virtual rays as solid lines: forgetting that rays extending behind a mirror (such as to the virtual focal point or virtual image in convex mirrors) must be drawn as dashed lines.
- Incorrect ray direction through \(C\): failing to show arrows pointing in both directions for rays passing through the centre of curvature \(C\) that reflect straight back along their incident path.
- Assuming convex mirrors can form real or magnified images: believing that changing the object distance can produce a real or magnified image in a convex mirror; in reality, convex mirror images are always virtual, upright, and diminished.
Concept Misunderstandings
All curved mirrors magnify objects.
Convex mirrors always diminish images (making them smaller than the object). Concave mirrors only magnify when the object is placed at \(f < u < 2f\) (inverted) or \(u < f\) (upright).
The focal point of a convex mirror is in front of the mirror where light hits.
For a convex mirror, parallel incident rays diverge after reflection. The focal point, \(F\), is located behind the mirror where the reflected rays appear to diverge from.
Plane mirrors have the widest field of vision because they do not distort shapes.
Convex mirrors have a wider field of vision than plane mirrors because the outward curvature allows light from a broader angular field to reach the observer's eye.
Summary
- Spherical mirrors are curved reflecting surfaces: concave (reflects on inside) and convex (reflects on outside).
- Mirror geometry is defined by the pole (\(P\)), focal point (\(F\)), centre of curvature (\(C\)), focal length (\(f\)), and radius of curvature (\(r\)), where \(r = 2f\).
- In concave mirrors, parallel rays converge at \(F\) (in front of mirror). In convex mirrors, parallel rays appear to diverge from \(F\) (behind mirror).
- Field of vision size comparison: Concave mirror (very narrow) < Plane mirror (narrow) < Convex mirror (wide).
- Concave mirror image characteristics depend on object distance \(u\): \(u > 2f\): Real, inverted, diminished (\(f < v < 2f\)); \(u = 2f\): Real, inverted, same size (\(v = 2f\)); \(f < u < 2f\): Real, inverted, magnified (\(v > 2f\)); \(u = f\): Virtual, upright, magnified (at infinity); \(u < f\): Virtual, upright, magnified (\(v > u\), behind mirror).
- Convex mirror image characteristics: Always virtual, upright, and diminished for all object distances (\(v < f\)).
- Real-world uses: Concave mirrors are used in cosmetic mirrors, dental mirrors, and headlights; convex mirrors are used in blind spot mirrors, security mirrors, and vehicle rear mirrors.
Short Activity
Objective: Identify image characteristics, fields of vision, and applications of spherical mirrors.
A. Objective Quiz
1 What are the characteristics of the image formed by a concave mirror when an object is placed between the focal point, \(F\), and the pole of the mirror, \(P\) (\(u < f\))?
2 Which type of mirror provides the widest field of vision for an observer?
B. Fill in the Blanks
3 The distance between the pole of the spherical mirror, \(P\), and the centre of curvature, \(C\), is known as the of curvature.
4 For a spherical mirror, the radius of curvature, \(r\), is related to its focal length, \(f\), by the formula \(r =\) .
C. Matching / Drag and Drop
Match each mirror application to its correct type of spherical mirror and purpose. If using a phone, tap the answer first, then tap the matching box.
