Mirrors are used in torches, vehicle headlights, rear-view mirrors, telescopes, shaving mirrors and dentists' instruments. This chapter studies how spherical mirrors form real and virtual images and derives the mirror formula connecting object distance, image distance and focal length. Throughout, the real-is-positive, virtual-is-negative sign convention is used, as in the NEB textbook.
Terms Used for Spherical Mirrors
A spherical mirror is a part of a hollow sphere whose one surface is polished. If the inner (hollow) surface reflects, it is a concave mirror; if the outer (bulging) surface reflects, it is a convex mirror.
Term
Meaning
Pole (P)
Geometric centre of the reflecting surface
Centre of curvature (C)
Centre of the sphere of which the mirror is a part
Radius of curvature (R)
Radius of that sphere, R = PC
Principal axis
Line joining P and C
Aperture
Diameter of the reflecting surface (the width M₁M₂)
Principal focus (F)
Point on the axis where rays parallel and close to the axis meet (concave) or appear to diverge from (convex) after reflection
Focal length (f)
Distance PF
Figure 14.1: Concave mirror (top) and convex mirror (bottom) with pole P, focus F, centre of curvature C, focal length f and radius R
Relation between focal length and radius of curvature
Consider a ray AB parallel to the principal axis (close to it) striking the mirror at B. The normal at B is the line CB, so the angle of incidence is ∠ABC = θ and the angle of reflection is ∠CBF = θ. Since AB ∥ PC, ∠BCP = θ (alternate angles). Hence triangle CFB has equal angles ∠FCB = ∠FBC = θ, so FC = FB.
For a small aperture, B is very close to P, so FB ≈ FP
FC = FP ⇒ PF = ½ PC
f = R / 2
14.1 Real and Virtual Images
14.1(a) Real and Virtual Objects
An object is real if the rays reaching the mirror actually diverge from it. It is virtual if the rays reaching the mirror are already converging towards a point behind the mirror. Ordinary objects placed in front of a mirror are real objects.
14.1(b) Real Image
An image is real when the reflected rays actually meet at a point. A real image can be caught on a screen and is always inverted in a mirror. It is formed on the same side of the mirror as the real object.
14.1(c) Virtual Image
An image is virtual when the reflected rays only appear to come from a point, because they actually diverge. Their backward extensions meet behind the mirror. A virtual image cannot be caught on a screen but can be seen by the eye (or a lens). It is always erect in a mirror.
Property
Real image
Virtual image
Rays
Actually converge
Appear to diverge from a point
Screen
Can be obtained
Cannot be obtained
Position
In front of the mirror
Behind the mirror
Orientation
Inverted
Erect
Sign of v (this convention)
Positive (+)
Negative (−)
14.1(d) Rules for Drawing Ray Diagrams
To locate the image of a point on an object, any two of the following rays are traced:
Ray 1: A ray parallel to the principal axis passes through F after reflection (concave), or appears to come from F (convex).
Ray 2: A ray through F (or heading towards F) becomes parallel to the principal axis after reflection.
Ray 3: A ray through C (or heading towards C) strikes the mirror normally and is reflected back along its own path.
Ray 4: A ray striking the pole P is reflected so that the angle of reflection equals the angle of incidence about the principal axis.
14.1(e) Images in a Concave Mirror
Figure 14.2: Object beyond C. Rays used: Ray 1 (parallel to the axis, then through F) and Ray 4 (incident at the pole P)
Figure 14.3: Object between F and P. Dashed lines are backward extensions of reflected rays; they meet behind the mirror at the virtual image
Object position (concave mirror)
Image position
Nature and size
At infinity
At F
Real, inverted, point-sized
Beyond C
Between F and C
Real, inverted, diminished
At C
At C
Real, inverted, same size
Between C and F
Beyond C
Real, inverted, magnified
At F
At infinity
Real, inverted, highly magnified
Between F and P
Behind the mirror
Virtual, erect, magnified
14.1(f) Images in a Convex Mirror
Figure 14.4: Image in a convex mirror. Reflected rays diverge; their backward extensions (dashed) meet behind the mirror
Uses of convex mirrors:
Rear-view mirrors of vehicles: erect image and a wide field of view.
Street and shop security mirrors: view of a large area.
Uses of concave mirrors:
Shaving and make-up mirrors, dentist's mirror: enlarged erect image at close range.
Headlights, torches, solar cookers: a source at F gives a parallel beam, or parallel sunlight is concentrated at F.
Reflecting telescopes: collect light from distant objects.
14.2 Mirror Formula
14.2(a) Sign Convention (Real is Positive, Virtual is Negative)
All distances are measured from the pole P. The convention is based on whether an object, an image or a focus is real or virtual:
Quantity
Positive (+)
Negative (−)
Object distance u
Real object
Virtual object
Image distance v
Real image (in front of mirror)
Virtual image (behind mirror)
Focal length f (and R)
Concave mirror (real focus)
Convex mirror (virtual focus)
An ordinary object placed in front of a mirror is real, so u is positive in almost every problem.
A concave mirror has f = +R/2; a convex mirror has f = −R/2.
If the calculated v is positive the image is real (inverted); if negative, it is virtual (erect).
Heights above the principal axis are taken as positive; the real inverted image has a downward height.
Figure 14.5: Sign convention: real quantities positive, virtual quantities negative
14.2(b) Derivation of the Mirror Formula
Figure 14.6: Geometry for deriving the mirror formula (concave mirror, real image). Triangles ABP and A′B′P are similar; so are DPF and A′B′F
Derivation of 1/f = 1/u + 1/v
Let AB be a small real object at distance u (= AP) from a concave mirror and A′B′ its real image at distance v (= A′P). Draw ray BP striking the pole, and ray BD parallel to the axis, which after reflection passes through F and meets BP's reflected ray at B′.
Step 1: Triangles ABP and A′B′P are similar (angle of incidence = angle of reflection at P):
A′B′ / AB = A′P / AP = v / u ... (1)
Step 2: Triangles DPF and A′B′F are similar, and DP = AB:
A′B′ / DP = A′F / PF
A′B′ / AB = (v − f) / f ... (2)
Step 3: Equate (1) and (2):
v / u = (v − f) / f
vf = uv − uf
uf + vf = uv
Dividing every term by uvf:
1/f = 1/u + 1/v
With the real-is-positive convention this single formula holds for concave and convex mirrors, real and virtual images; only the signs of f, u and v change.
14.2(c) Linear Magnification
Linear magnification m is the ratio of the height of the image (I) to the height of the object (O). From equation (1):
Substituting a positive f for a convex mirror. A convex mirror always has negative f.
Forgetting that a negative v means the image is virtual, not that the distance is "wrong".
Using 1/f = 1/u − 1/v: that belongs to a different convention. Here it is always +.
Applying the formula to large-aperture mirrors: it holds only for rays close to the axis (paraxial rays).
Example 14.1 — Real image
An object is placed 30 cm from a concave mirror of focal length 15 cm. Find the image position and magnification.
f = +15 cm, u = +30 cm
1/v = 1/f − 1/u = 1/15 − 1/30 = 1/30
v = +30 cm (real, in front of the mirror)
m = −v/u = −30/30 = −1 (inverted, same size)
The object is at C (R = 30 cm), so the image is also at C.
Example 14.2 — Virtual image in a concave mirror
A 2 cm tall object is placed 6 cm from a concave mirror of focal length 10 cm. Find the image position and size.
1/v = 1/10 − 1/6 = (3 − 5)/30 = −1/15
v = −15 cm (virtual, behind the mirror)
m = −v/u = 15/6 = +2.5 (erect)
Image height = 2.5 × 2 = 5 cm
Example 14.3 — Convex mirror
An object is placed 30 cm from a convex mirror of focal length 20 cm. Find the image position and magnification.
f = −20 cm (virtual focus), u = +30 cm
1/v = 1/f − 1/u = −1/20 − 1/30 = −5/60 = −1/12
v = −12 cm (virtual, behind the mirror)
m = −v/u = 12/30 = +0.4 (erect, diminished)
Example 14.4 — Finding f and R
A real image forms 60 cm from a concave mirror when the object is 20 cm away. Find f and R.
1/f = 1/u + 1/v = 1/20 + 1/60 = 4/60
f = 15 cm
R = 2f = 30 cm
Example 14.5 — Magnified image on a screen
A concave mirror forms a 4 times magnified real image on a screen 40 cm from the mirror. Find the object distance and focal length.
|m| = v/u = 4 ⇒ u = 40/4 = 10 cm
1/f = 1/10 + 1/40 = 5/40 = 1/8
f = 8 cm
Key Formulas and Summary
Fundamental Relations:
f = R / 2
1/f = 1/u + 1/v
m = I/O = − v/u
Sign Convention (real +, virtual −):
Real object: u > 0 Virtual object: u < 0
Real image: v > 0 Virtual image: v < 0
Concave mirror: f > 0 Convex mirror: f < 0
Quick Facts:
Real image: can be on a screen, inverted, v > 0.
Virtual image: cannot be on a screen, erect, v < 0.
Convex mirror: always virtual, erect and diminished for a real object.
Concave mirror: virtual image only when u < f.
Practice Numerical Problems
Use 1/f = 1/u + 1/v with real-is-positive signs. Attempt each problem before reading the answer.
1. A concave mirror has f = 12 cm. An object is placed 18 cm in front of it. Find the image distance and magnification.
Answer: 1/v = 1/12 − 1/18 = 1/36, so v = +36 cm (real); m = −36/18 = −2 (inverted, magnified).
2. A concave mirror has radius of curvature 40 cm. Find the image for an object 10 cm from it.
Answer: f = 20 cm. 1/v = 1/20 − 1/10 = −1/20, so v = −20 cm (virtual); m = +2 (erect, magnified).
3. A convex mirror has f = 15 cm (magnitude). An object is 25 cm from it. Find the image position and magnification.
Answer: f = −15 cm. 1/v = −1/15 − 1/25 = −8/75, so v ≈ −9.4 cm (virtual); m = 9.375/25 ≈ +0.375.
4. An erect image 3 times the size of the object is obtained with a concave mirror when the object is 10 cm from it. Find the focal length.
Answer: Erect means virtual: m = −v/u = +3 ⇒ v = −30 cm. 1/f = 1/10 − 1/30 = 2/30, so f = 15 cm.
5. A car's rear-view convex mirror has radius of curvature 2 m. A vehicle is 5 m behind. Find the image position and magnification.
Answer: f = −1 m. 1/v = −1 − 1/5 = −6/5, so v ≈ −0.83 m (virtual); m ≈ +0.17.
6. A real image is formed at the same distance (50 cm) as the object from a concave mirror. Find f and R.
Answer: 1/f = 1/50 + 1/50 = 2/50, so f = 25 cm, R = 50 cm. The object is at C.
7. A 4 cm tall object is 15 cm from a concave mirror of f = 10 cm. Find the position, nature and height of the image.
Answer: 1/v = 1/10 − 1/15 = 1/30, so v = +30 cm (real); m = −2; height = 8 cm, inverted.
8. Where should an object be placed in front of a concave mirror of f = 20 cm to get a virtual image 40 cm behind the mirror?
Answer: v = −40 cm. 1/u = 1/f − 1/v = 1/20 + 1/40 = 3/40, so u ≈ 13.3 cm; m = +3.
Conceptual and Long Answer Questions
1. Differentiate between a real and a virtual image.
See table in 14.1: a real image is formed by actual convergence of rays, can be received on a screen and is inverted; a virtual image only appears to come from behind the mirror, cannot be received on a screen and is erect.
2. Why is a convex mirror preferred as a rear-view mirror?
It always forms an erect, diminished image, which gives a wide field of view.
3. Why does a dentist use a concave mirror?
With the tooth between F and P, a virtual, erect and magnified image is formed.
4. Can a real image be formed by a convex mirror of a real object? Explain.
No. For a real object, reflected rays always diverge, so the image is always virtual.
5. Prove that f = R/2 for a spherical mirror. (Section 14, introduction)
6. Derive the mirror formula 1/f = 1/u + 1/v for a concave mirror using the real-is-positive convention. (Section 14.2)
Multiple Choice Questions
1. A real image is one that:
(a) Cannot be obtained on a screen
(b) Is formed by actual meeting of reflected rays ✓
(c) Is always erect
(d) Is formed behind a mirror
2. In the real-is-positive convention, the focal length of a concave mirror is:
(a) Positive ✓
(b) Negative
(c) Zero
(d) Infinite
3. The focal length of a spherical mirror is related to its radius of curvature by:
(a) f = R
(b) f = 2R
(c) f = R/2 ✓
(d) f = R/4
4. A convex mirror always forms an image that is:
(a) Real, inverted, diminished
(b) Real, erect, magnified
(c) Virtual, erect, diminished ✓
(d) Virtual, inverted, magnified
5. An object is at the centre of curvature of a concave mirror. The image is:
(a) At infinity
(b) At F, diminished
(c) At C, real, inverted, same size ✓
(d) Behind the mirror
6. A calculated image distance v = −24 cm means the image is:
(a) Real, in front of the mirror
(b) Virtual, behind the mirror ✓
(c) At infinity
(d) Inverted
7. The mirror formula in the real-is-positive convention is:
(a) 1/f = 1/v − 1/u
(b) 1/f = 1/u + 1/v ✓
(c) f = u + v
(d) 1/f = u + v
8. When an object is placed between F and P of a concave mirror, the image is:
Physics Decoded
Grade XI › Reflection at Curved Mirrors
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