26.1 CURVED MIRRORS
26.1.1 Spherical Mirrors
1 The shape of concave and convex mirrors
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| Figure 26.1 illustrates spherical mirrors as parts of an imaginary sphere. (a) A concave mirror, where the reflecting surface curves inward. (b) A convex mirror, where the reflecting surface curves outward. |
The reflecting surface of a spherical mirror can be thought of as part of an imaginary sphere. Figure 26.1(a) shows a concave mirror and Figure 26.1(b) shows a convex mirror as part of a sphere.
2 Some terms concerning curved mirrors
(a) Centre of curvature
The centre C of the imaginary sphere is known as the centre of curvature of the mirror.
(b) Radius of curvature
The radius r of the imaginary sphere is known as the radius of curvature of the mirror.
(c) Pole
The point P at the centre of the mirror is known as the pole.
(d) Principal axis
The line which passes through C and P is known as the principal axis of the mirror.
26.1.2 Focus and Focal Length
1 Focus
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| Figure 26.2 shows how light rays behave in spherical mirrors. (a) In a concave mirror, parallel rays of light are reflected and converge at a single point F in front of the mirror. This point is called the focus. (b) In a convex mirror, parallel rays diverge after reflection and appear to originate from a point F behind the mirror. This point is the focus of the mirror. |
A narrow parallel beam of light along the principal axis of a curved mirror reaches the reflecting surface of the mirror. Consider the following mirrors:
(a) Concave mirror
(i) The reflected light converges towards one unique point F in front of the mirror, as shown in Figure 26.2(a). This point is known as the focus.
(ii) The focus is in front of a concave mirror.
(b) Convex mirror
(i) The reflected light diverges from the mirror. The ray appears to originate from a point F behind the mirror, as shown in Figure 26.2(b). This point is the focus of the mirror.
(ii) The focus is behind the mirror.
2 Focal length
The length f between F and P is known as the focal length.
26.1.3 Relationship between f and r
We can show that for a spherical concave mirror and convex mirror, the focal length f of the mirror is related to the radius of curvature r by the following equation:
f = 1/2 r
26.1.4 Images Produced by Curved Mirrors
1 Real image
Light reflected by a curved mirror can form an image of the object. The image is a real image if it can be formed on a screen. Light rays actually pass through a real image.
2 Virtual image
If the image cannot be formed on a screen, it is a virtual image. No light rays pass through a virtual image.
26.2 RAY DIAGRAM
26.2.1 How to Draw a Ray Diagram
To draw a ray diagram of a given curved mirror, we normally need to use two of the three ‘special’ rays. Consider the following steps:
Ray 1:
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| Figure 26.3 shows how light rays behave in spherical mirrors. (a) In a concave mirror, parallel rays of light are reflected and converge at a single point F in front of the mirror. This point is called the focus. (b) In a convex mirror, parallel rays diverge after reflection and appear to originate from a point F behind the mirror. This point is the focus of the mirror. |
(a) From the tip of the object draw a ray parallel to the principal axis towards the mirror.
(b) For a concave mirror, let the reflected ray pass through the focus F, as shown in Figure 26.3(a).
(c) For a convex mirror, draw the reflected ray as if it originated from the focus, as shown in Figure 26.3(b).
Ray 2:
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| Figure 26.4 illustrates a ray drawn from the tip of the object through the centre of curvature (C) of a spherical mirror. Since the ray strikes the mirror perpendicular to its surface, it is reflected back along the same path. This behavior is shown for both (a) concave mirror and (b) convex mirror. |
From the tip of the object draw a ray passing through the centre of curvature C. The ray is perpendicular to the surface of the mirror. Because of this, the reflected ray will travel along its original path, as shown in Figure 26.4(a),(b).
Ray 3:
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| Figure 26.5 shows a ray drawn from the tip of the object passing through the focus (F) of a spherical mirror. After reflection, the ray travels parallel to the principal axis. This behavior is illustrated for both (a) concave mirror and (b) convex mirror. |
From the tip of the object draw a ray passing through the focus F. Then draw the reflected ray parallel to the principal axis, as shown in Figure 26.5(a),(b).
26.2.1 Using Ray Diagram to Deduce the Nature of Image Formed by Concave Mirror
A ray diagram can be drawn to show how the image of an object is formed by light reflected by a concave mirror. From this diagram, we can deduce the characteristics of the image.
Consider how images of the same object are formed when the object is placed in the following positions in front of the concave mirror:
(a) Object distance longer than 2f
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| Figure 26.6 shows the formation of an image by a concave mirror when the object is placed in an upright position. From the ray diagram, the image formed is inverted (upside down), real, and diminished (smaller than the object). The image is formed at a point between the centre of curvature (C) and the pole (P) in front of the mirror. |
Refer to Figure 26.6. Normally the object is drawn in the upright position. From the ray diagram, we deduce that the image is
(i) inverted, i.e., upside down
(ii) real
(iii) diminished, i.e., size smaller than that of the object
(iv) formed at a point between C and P in front of the mirror
(b) Object distance equals 2f
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| Figure 26.7 shows the image formed by a concave mirror when the object is placed at the centre of curvature (C). The image formed is inverted, real, and of the same size as the object. It is located at the centre of curvature (C) in front of the mirror. |
Refer to Figure 26.7. The image is
(i) inverted
(ii) real
(iii) same size as that of the object
(iv) formed at C in front of the mirror
(c) Object distance between f and 2f
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| Figure 26.8 shows the image formed by a concave mirror when the object is placed between the centre of curvature (C) and the focus (F). The image formed is inverted, real, and magnified (larger than the object). It is formed at a distance greater than CP in front of the mirror. |
Refer to Figure 26.8. The image is
(i) inverted
(ii) real
(iii) magnified, i.e., size larger than that of the object
(iv) formed at a distance longer than CP in front of the mirror
(d) Object distance less than f
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| Figure 26.9 shows the image formed by a concave mirror when the object is placed between the focus (F) and the pole (P). The image formed is upright, virtual, and magnified. It is formed behind the mirror. |
Refer to Figure 26.9. The image is
(i) upright
(ii) virtual
(iii) magnified
(iv) formed behind the mirror
Take note of the following:
If the object distance is
(a) longer than f, the image is inverted, real and formed in front of the mirror
(b) less than f, the image is upright, virtual, magnified and formed behind the mirror
26.2.3 Using Ray Diagram to Deduce Nature of Image Formed by Convex Mirror
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Figure 26.10 shows that regardless of the object distance, a convex mirror always forms an image that is upright, virtual, and diminished (smaller than the object). The image is always formed behind the mirror. |
Refer to Figure 26.10. Irrespective of any object distance, the image of a real object formed by a convex mirror is always
(a) upright
(b) virtual
(c) diminished
(d) formed behind the mirror.
Note: A convex mirror can form a real image but we need to use a ‘virtual object’.
EXAMPLE 26.1
A spherical concave mirror has a radius of curvature of 20 cm. State the characteristics of the image produced by the mirror of an object placed
(i) 5.0 cm
(ii) 15 cm
(iii) 30 cm away from the mirror.
Answer
r = 20 cm
f = 1/2 r
= 1/2 (20) = 10 cm
| Object Distance | Orientation | Type | Size | Position |
|---|---|---|---|---|
| Less than f | Upright | Virtual | Magnified | Behind mirror |
| Between f and 2f | Inverted | Real | Magnified | In front |
| Longer than 2f | Inverted | Real | Diminished | In front |
Applications of Curved Mirrors
Curved mirrors, including concave and convex mirrors, are widely used in everyday life and technology. Their ability to reflect light and form images makes them essential in many practical applications.
Makeup & Shaving Mirrors
Concave mirrors are used in makeup and shaving mirrors because they produce magnified and upright images when the object is placed close to the mirror.
Vehicle Side Mirrors
Convex mirrors are used as side mirrors in vehicles because they provide a wider field of view and help drivers see more area behind them.
Dental Mirrors
Dentists use concave mirrors to obtain enlarged images of teeth, making it easier to examine small details inside the mouth.
Headlights & Torches
Concave mirrors are used in headlights and flashlights to produce parallel beams of light, improving visibility over long distances.
Security Mirrors
Convex mirrors are used in shops, parking areas, and road corners to provide a wide-angle view and improve safety.
Solar Cookers
Concave mirrors focus sunlight to a single point, generating heat that can be used for cooking and energy applications.
Conclusion
Curved mirrors, including concave and convex mirrors, play an essential role in optics and everyday applications. Their ability to reflect light and form different types of images makes them highly useful in both science and technology.
By understanding ray diagrams, focal length, and image formation, we can predict how images are formed under different object positions. Concave mirrors can produce real or virtual images depending on distance, while convex mirrors always produce virtual and diminished images.
In conclusion, curved mirrors are not only important in physics but also widely used in practical applications such as vehicle mirrors, medical instruments, and optical devices, making them a key concept in understanding light and reflection.
Frequently Asked Questions (Curved Mirrors)
Curved mirrors are mirrors with a curved reflecting surface, including concave and convex mirrors, used to form images by reflection of light.
Concave mirrors curve inward and can form real or virtual images, while convex mirrors curve outward and always form virtual, upright, and diminished images.
The focal point is the point where parallel rays of light converge (concave) or appear to diverge from (convex) after reflection.
Focal length is the distance between the pole (P) and the focus (F) of a curved mirror.
The focal length is half of the radius of curvature, meaning f = r/2.
A real image is formed when reflected rays actually meet and can be projected onto a screen.
A virtual image is formed when reflected rays do not actually meet and cannot be projected onto a screen.
Convex mirrors spread light rays outward, allowing a larger area to be seen, which is useful for safety and surveillance.
Ray diagrams are used to determine the position, size, and nature of images formed by mirrors.
Applications include vehicle mirrors, makeup mirrors, dental mirrors, headlights, and security mirrors.
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