How to derive lens maker's formula ? give simplest method

Asked by Sumit Joshi | 31st May, 2013, 08:19: AM

Expert Answer:

Figure (a)

Figure (b)

Figure (c)

The above figure shows the image formation by a convex lens.

Assumptions made in the derivation:

  • The lens is thin so that distances measured from the poles of its surfaces can be taken as equal to the distances from the optical centre of the lens.

  • The aperture of the lens is small.

  • The object consists only of a point lying on the principle axis of the lens.

  • The incident ray and refracted ray make small angles with the principle axis of the lens.

A convex lens is made up of two convex spherical refracting surfaces.

The first refracting surface forms image I of the object O [figure (b)].

Image I1 acts as virtual object for the second surface that forms the image at I [figure (c)]. Applying the equation for spherical refracting surface to the first interface ABC, we obtain

A similar procedure applied to the second interface ADC gives

For a thin lens, BI1 = DI1

Adding equations (i) and (ii), we obtain

Suppose the object is at infinity i.e.,

OB ? ? and DI ? f

Equation (iii) gives

The point where image of an object placed at infinity is formed is called the focus (F) of the lens and the distance f gives its focal length. A lens has two foci, F and, on either side of it by the sign convention.

BC1 = R 1

CD2 = ?R 2

Therefore, equation (iv) can be written as

Equation (v) is known as the lens maker’s formula.

From equations (iii) and (iv), we obtain

As B and D both are close to the optical centre of the lens,

BO = ? u, DI = + v, we obtain

Equation (vii) is known as thin lens formula.

Answered by  | 31st May, 2013, 08:22: PM

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