Question

Derive the lens maker’s formula.

Answer 1

For refraction at LP₁N,

`mu_1/(CO) + mu_2/(CI_1) = (mu_2 - mu_1)/(C C_1)`

As if the image is formed in the denser medium.

For refraction at LP₂N

`mu_2/(- CI_1) + mu_1/(CI) = (mu_2 - mu_1)/(C C_2)`

As if the object is in the denser medium and the image is formed in the rarer medium.

Combining the refractions at both the surfaces.

`mu_1/(CO) + mu_2/(CI) = mu_2 - mu_1 (1/(C C_1) + 1/(C C_2))`

Substituting the values with sign conventions,

`1/-u + 1/v = (mu_2 - mu_1)/mu_1 (1/R_1 - 1/R_2)`

Since `mu_2/mu_1` = µ

`1/-u + 1/v = (mu_2 - mu_1)/mu_1 (1/R_1 - 1/R_2)`

OR

`1/-u + 1/v = (mu - 1) (1/R_1 - 1/R_2)`

When the object is kept at infinity, the image is formed at the principal focus.

i.e., u = −∞, v = +f

`1/f = (mu - 1) (1/R_1 - 1/R_2)`

This equation is called ‘Lens Maker’s Formula’.