Question

(a) Draw a ray diagram for formation of image of a point object by a thin double convex lens having radii of curvatures R₁ and R₂ and hence derive lens maker’s formula. 

(b) Define power of a lens and give its S.I. units. 

If a convex lens of focal length 50cm is placed in contact coaxially with a concave lens of focal length 20cm, what is the power of the combination?

Answer 1

(a) 

From lens maker formula,

\(\frac 1f = (\mu - 1) \left(\frac 1{R_1} - \frac 1{R_2}\right)\)

Assumptions

The following assumptions are taken for the derivation of lens maker formula.

• Let us consider the thin lens shown in the image above with 2 refracting surfaces having the radii of curvatures R₁ and R₂ respectively. 

• Let the refractive indices of the surrounding medium and the lens material be n₁ and n₂ respectively.

Derivation

The complete derivation of lens maker formula is described below. Using the formula for refraction at a single spherical surface we can say that,

For the first surface,

\(\frac{n_2}{v_1} - \frac{n_1}u = \frac{n_2 - n_1}{R_1} \)   .......(1)

For the second surface,

\(\frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1 - n_2}{R_2} \)    .....(2)

Now adding equation (1) and (2),

\(\frac {n_1}v - \frac {n_1}u = \left({n_2}-{n_1} \right) \left[\frac 1{R_1} - \frac 1{R_2}\right]\)

⇒ \(\frac 1v - \frac 1u = \left(\frac{n_2}{n_1} - 1\right) \left[\frac 1{R_1} - \frac 1{R_2}\right]\)

When u = \(\infty\) and v = f

\(\frac 1f = \left(\frac{n_2}{n_1} - 1\right) \left[\frac 1{R_1} - \frac 1{R_2}\right]\)

But also,

\(\frac 1v - \frac 1u = \frac 1f\)

Therefore we can say that,

\(\frac 1f = (u - 1) \left(\frac 1{R_1} - \frac 1{R_2}\right)\)

Where µ is the refractive index of the material.

This is the lens maker formula derivation.

(b) Power of a lens is its ability to converge or diverge the rays of light falling on it. Power of a lens is equal to reciprocal of the focal length of the lens.

SI unit of power is dioptre (D).

Given the focal length of convex lens, f₁ = 50 cm = 0.5 m

The focal length of concave lens, f₂ = 20 cm = 0.2 m

The equivalent focal length of the combination of lenses is given by the sum of individual focal lengths.

\(\frac 1{f_c} = \frac 1{f_1} + \frac 1{f_2}\)

Since the focal length of concave lens is negative, hence

\(\frac 1{f_c} = \frac 1{f_1}- \frac 1{f_2}\)

Substituting the values,

\(\frac 1{f_c} = \frac 1{0.5}- \frac 1{0.2}\)

\(= \frac{0.2 - 0.5}{0.5 \times 0.2}\)

⇒ \(\frac 1{f_c} = \frac{-0.3}{0.1} = -3\)

Thus, the power of the combination is

\(P_c = \frac 1{f_c } = -3D\)

Hence, the power of the combination of the lenses is -3D.

Answer 2

(a) Lens Maker’s Formula: Suppose L is a thin lens. The refractive index of the material of lens is n₂ and it is placed in a medium of refractive index n₁. The optical centre of lens is C and X ' X is principal axis. The radii of curvature of the surfaces of the lens are R₁  and R₂  and their poles are P₁  and P₂. The thickness of lens is t, which is very small. O is a point object on the principal axis of the lens. The distance of O from pole P₁  is u. The first refracting surface forms the image of O at I' at a distance v' from P₁.From the refraction formula at spherical surface

The image I' acts as a virtual object for second surface and after refraction at second surface, the final image is formed at I. The distance of I from pole P₂  of second surface is v. The distance of virtual object (I') from pole P₂  is (v' - t). 

For refraction at second surface, the ray is going from second medium (refractive index n₂ ) to first medium (refractive index n₁ ), therefore from refraction formula at spherical surface

For a thin lens t is negligible as compared to v', therefore from (ii)

Adding equations (i) and (iii), we get 

where ₁n₂ = n₂/n₁ = is refractive index of second medium (i.e. medium of lens) with respect to first medium. 

If the object O is at infinity, the image will be formed at second focus i.e. if u= ∞, v = f₂ = f

Therefore from equation (iv)

This is the formula of refraction for a thin lens. This formula is called Lens-Maker’s formula. 

If first medium is air and refractive index of material of lens be n, then ₁n₂ = n, therefore equation (v) may be written as

(b) Power of a Lens: The power of a lens is its ability to deviate the rays towards its principal axis. It is defined as the reciprocal of focal length in metres.