What is the power of lens derive the power of combination of two lenses which are in contact?

The reciprocal of the focal length of a lens in metres is called the power of the lens.

The capacity of a lens to bend the rays of light depends upon the focal length. The smaller the focal length, the greater is the bending of a ray of light and vice- versa. Thus the power of a lens to bend the rays of light is inversely proportional to the focal length of the lens.

Units of power of a lens : The units of the power of a lens are dioptres.
One dioptre is the power of a lens of one metre focal length. The power of a lens is denoted by D. It is this dioptric number which the doctors prescribe for the spectacles of a person.

Mathematically, D = 1/Focal length in metre

= 100/Focal length in centimetre

The power of a convex lens is positive and that of a concave lens is negative.
Thus a convex lens of focal length 40 cm is said to possess a power of + 2.5 dioptre. similarly the power of a concave lens of focal length 20 cm is 5.0 dioptre.

Power of combination of lenses : The focal length of the combination of two lenses placed in contact is given by the relation.

   1/F = 1/f1 +1/f2

If p1 is the power of lens of focal length f1, p2 the power of the focal length f2 and P the power of the combination of focal length, F then,

P = p1 +p2

Thus the power of combination of lenses is the algebraic sum of the powers of individual lenses.
If two lenses are separated by distance d metre, then power of combination of lenses is given by relation

P = P1 + P2 – d∙(P1∙P2)

Practice Problems :

1. A lens is formed by combining two thin lenses of powers + 12 D and – 8 D in contact with each other. What will be the focal length of combination ?
Solution:
P1 = + 12 D, P2 = – 8 D
Power of combination (P) = P1 + P2
= 12 – 8
= 4 D
Focal length of combination (f) = 1/P = 1/+4
= 0.25 m
= 25 cm. (Ans.)

2. An optician prescribes spectacles to a patient with a combination of a convex lens of focal length 40 cm and a concave lens of 25 cm focal length. What will be the power of spectacles ?
Solution:
F1 = 40 cm = 0.4 m
F2 = 25 cm = 0.25 m

P1 = 1/F1 = 1/0.4 = 2.5 D
P2 = 1/F2 = – 1/F2 = –1/0.25 = –4∙0 D {(Because for concave mirror, P = –1/F)

Power of spectacles
(P) = P1 + P2
= 2∙5 + (–4∙0)
= – 1∙5 D (Ans.)

3. A convex lens of power +4 D and a concave lens of power 3 D are placed in contact. What is equivalent power of the combination ?
Solution:
P1 = 4 D,
P2 = –3 D,
P = P1 + P2
= 4D – 3D
= 1D
Equivalent power of combination is 1 Dioptre.

4. A convex lens has focal length of 20 cm. find its power in Dioptres.
Solution:
F = 20 cm
= 0∙2 m
P = 1/F
= 1/0∙2
= +5
Power of convex lens is +ve and in magnitude it is 5 Dioptre.

5. The combined power of two lenses in contact is + 10 D. When they are separated by 20 cm, their power becomes 6.25 D. Find the power of individual lenses.
Solution:
Given P = P1 + P2 = +10 D                              ------(1)
Distance between lenses, d = 20 cm = 0∙2 m
When these lenses are separated by distance 0∙2 m,

Power of combination becomes
P' = P1 +P2 – d∙(P1∙P2) = 6∙25
= 10 – 0∙2(P1∙P2) = 6∙25

P1∙P2 = (10 – 6∙25) / 0∙2
= 18∙75

(P1 – P2)2 = (P1 + P2)2 – 4∙(P1∙P2)

=         (10)2 – 4∙(18∙75)
(P1 – P2)2  = 25
P1 – P2 = 5 D                                     ------(2)

Adding equations (1) and (2)
2P1 = 15D
or P1 = 15/2 = 7∙5 D

Subtracting equation (2) from (1)
2P2   = 10 – 5 = 5
P2 = 5/2 = 2∙5 D

Hence power of lenses, P1 = 7∙5 D and P2 = 2.5 D.

6. A lens made of glass of refractive index 1∙5 has power of + 10 D when placed in air, find the power of same lens when immersed in water (µ = 1∙33). Also find the change in power of lens.
Solution:
When lens is immersed in water, the focal length of lens increases by nearly 4 times as when in air.

∴ Power of lens in water = Power of lens in air/4 = 10/4 = +2∙5 D
∴ Change in Power = + 2∙5 – 10∙0
= –7∙5 D (Nearly) Ans.

Some Multiple Objective Type Questions for Practice :

1. An optician prescribes spectacles to a patient with a combination of a convex lens of focal length 40 cm and concave lens of 25 cm. The power of spectacles is (a) 6∙0 D (b) 1∙5 D
(c) – 6∙0 D (d) – 1∙5 D

2. A convex lens of power +6 D is placed in contact with a concave lens of power –4 D. What will be the nature and focal length of this combination ?
(a) concave, 25 cm (b) convex, 50 cm
(c) Concave, 20 cm (d) convex, 100cm

3. A lens is formed combining two thin lenses in contact having power +12 D and – 8 D. The focal length of the combination is
(a) 25 cm (b) –25 cm
(c) 5 cm (d) – 5 cm

4. Two thin lenses of focal lengths 60 cm and –20 cm are placed in contact . The focal length of combination:
(a) 15 cm (b) 30 cm
(c) –15 cm (d) –30 cm

5. A convex lens has a focal length of 20 cm. Its power in dioptres is:
(a) –5 D (b) –10 D
(c) +5 D (d) +10 D

6. A convex lens of power +6 D is placed in contact with a concave lens of power –4D. What will be the nature and focal length of this combination ?
(a) convex, 100 cm (b) convex, 50 cm
(c) concave, 25 cm (d) concave, 20 cm

7. Convex lens of power 4D and a concave lens of power 3D are placed in contact. What is the equivalent power of the combination ?
(a) 7 D (b) 4/3 D
(c) 1 D (d) None of the above

8. A convex lens has a focal length of 20 cm. Its power in dioptres is
(a) –5 D (b) – 10 D
(c) +5D   (d) +10 D

9. The focal length of convex lens is 50 cm. What is its power ?
(a) +2 D (b) – 2D
(c) –50 D (d) +50 D

10. Two converging lenses of equal focal lengths f are placed in contact. the focal length of the combination is
(a) f (b) 2f
(c) f/2 (d) 3f

11. Two lenses one convex and other concave of focal length 0∙5 m and 1∙0 m respectively recombined, the power of combination will be
(a) –1∙0 D (b) +1∙0 D
(c) 0∙5 D (d) –0∙5 D

12. Two lenses of powers P1 = +2 D and P2 = –2 D respectively, are placed in contact. the power of combination is

(a) –2 D (b) +2 D
(c) +4 D (d) None of above

13. The power of combination of two lenses separated by distance d
(a) decreases as their separation is increased
(b) increases as their separation is increased
(c) remains same
(d) becomes equal to the distance 'd'

14. Two lenses of powers 6D and –5D are in contact with each other. The focal length of combination will be
(a) 1 cm (b) 20 cm
(c) 100 cm (d) 100/11 cm

15. Two thin lenses are in contact and the focal length of combination is 80 cm. If focal length of one of the lenses is 20 cm, then power of the other lens is
(a) +6∙0 D (b) –3∙75 D
(c) + 4∙0 D (d) –1∙5 D

16. A convex lens of power +6 D is placed in contact with a concave lens of power –4 D. The combination will act as a convex lens of focal length
(a) 2 cm (b) 20 cm
(c) 50 cm (d) 100 cm

Answers:
1. d
2. b
3. a
4. d
5. c
6. b
7. c
8. c
9. a
10.c
11.b
12.d
13.a
14.c
15.b
16.c