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LENS MAKER'S FORMULA

• IrJ2-1 -- J I s -+ tie -----

It------------r- M-U--f-~.Q -VB I

I; 'Y72-c. Y)j 'Y?2/' i: Y'2:'11 z, /~---------------rv~--,,-------- PR~~~-~-------

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Lens Equation - Law of distances part 11/04

Case (j) Convex lens

The real image of an object AB is formed as A'B' as shown in the figure below:

~A'B'C and ilABC are similar. Hence, AfBf =AfC -(1)AB AC

•• • AfBf AfFAlso, M'B'F and LlCLF-are similar.Hence, CL = CF - - - -(2)

But CL = AB.A'C A'F

SO from (1) and (2) we get - =-, AC CF Le

Cross multiplying we get vf=uv-uf

Dividing throughout by uvf,1 1 1-=---u f v

By sign convention u is nagative, v & R positive.

Thus1 1 1f = ;--~ I

Magnificaton (ro): It is the ratio of the height of the image to the height of the object. m;: hi/no

Use equation (1) with sign convention, we get -hi =..!..ho -u

hi vm=-=-ho u

Case tnlConcave lens -(Try yourself as explained above)

B.. li,/Object~

• I I

2F A F tluage Ci •2F'

Power of a lensThe power P of a lens is defined as the

tangent of the angle by which it converges or diverges abeam of light falling at unit distant from the optical centre.

hThus tan 8 = - put h = 1 and tan () l':; 8

11 1

8 = f or power(P) =,Hence power of a lens is the measure of reciprocal of focal length expressed in meters. Its unitis Dioptre(D)

Combination of lensesConsider two lenses Ll and L1:having focal lengths fl and f21 kept in contact.

o~-- - t.f -.--\-'7'

---

-----------~

The image formed by II at I' acts as the virtual object for the lens l2 and

hence a real image is formed at I as shown above.

1 1 1- = - - - -----------(1)h VI UForl21 1 1 1- = - - - -----------(2)fz 17 VI

Adding equations (1) and (2)1 1 1 1-+-=---11 fz V U

------------------- (3)

The combination can be regardes as a single lens of focal length If such that the positions of object and

image remain the same. Such a lens is called an equivalent lens.

Thus -------------( 4}

Comparing-equations (3) and (4) we get f!. = .!.+ .!.11 fz

~ 111It can be extended to many lenses as - = - +- + - +I It fz 13

Hence equivalent power of the combination is P = P1 + P2 + P3 + .

Defects of lenses

a. Spherical abberration.

It is the failure of a lens of large aperture to focus all the light rays to a single point so

that the final image is blurred. This can be eliminated by 'stops' or using suitablecombination of lenses.

f

b. Chromatic abberration

It is the failure of a thick lens to -converge all colours t-o a single point so that the finalimage is coloured and blurred.

Wh~

This can be eliminated by using a combination of convex and concave lens, called an'achromatic doublet', satisfying the condition,

Where COl and CO2 are the dispersive powers and f 1 and f21the focal lengths of the given lenses.

REFRACTION THROUGH A PRISM

let PQRSbe the course of a ray of light I passingA

through a triangular glass prism ABC of refracting

angle A asshown. The angle between the

In quadriiateral.AQNR,

directions of incident and emergent

rays is called the angle of

deviation, 6.s

LAQN = LARN = 90° B cHence A + N = 90° --------:--(1)

In~QNR, Comparing equations (1) & (2) we get,

A = rl + r2 ----------(3)

Further, Jn~QMR, 6 = .(i -n) + (e -rl.)

6 = i+ e - A ----·-14) .For different angles of incidence, corresponding angles of deviation are measured and a graph

is plotted between them as shown. It can be seen that, as 6

the angle of incidence Increases, theangleof deviation decreases 0 ....._.V....,to a minimum value and then increases. This minimum value

I

I

I

I

of deviation is called the angle of minimum deviation (Om)

At this angle, the refracted ray insldetheprlsm becomes parallel to

the base of the prism. At 0 = Dm, ri= r2 hence eqn. (3) becomes A = 2r so, r = A/2i = e hence eqn. (4) becomesu., = 2i - A so, i = (A + DmV2

B S II' I R f . . d Sin iy ne saw, e ractlve In ex, n = -. -Sinr

Sin{A + Dm)/2n= A

Sin{z)i.e.

For a small angled prism, it can be shown that 0 = A (n·1)