The lens maker’s equation is another formula used for lenses that gives us a relationship between the focal length, refractive index and radii of curvature of the two spheres used in lenses.

The lens maker’s equation for thin lenses is as given below as;

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

where,

f is the focal length (half the radius of curvature)

n is the refractive index of the material used

R1 is the radius of curvature of sphere 1

R2 is the radius of curvature of sphere 2

As previously mentioned, this equation can be used only for thin lenses. Lenses where the thickness is lesser, such that they are considered negligible in comparison to the radius of curvature, are referred to as thin lenses.

If the thickness of lens has to be considered in comparison to the radius of curvature, the following lens formula for thick lenses can be used.

\( \frac 1f ~=~(n-1) \left[ \frac {1}{R_1}~-~\frac{1}{R_2}~+~ \frac {( n-1) d}{n \times R_1 \times R_2} \right] \)

where,

d is the thickness of the lens in consideration.

Another situation which has to be considered is the lens maker’s formula accounting for objects that are present in different media. The equation is as follows,

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

where,

n1 is the refractive index of the lens in consideration

n2 is the refractive index of the medium outside

Have you asked yourself this question? “**What is the significance of ignoring the thickness sometimes and considering it sometimes?”**

Well, the answer to that is when the light ray travels from a medium of, say air, to the medium of the lens, it undergoes refraction. But when it travels from the medium of the lens to that of the air, it undergoes refraction again. The double refraction can be ignored if the lenses in optics is thin enough to make the assumption that light is refracted only once. This is done to make ray optics calculations simpler, but the first step would be to identify what constitutes thin and thick lenses.

## Lens maker’s formula limitations:

- The lens needs to be thin. This is because the separation between the two refracting surface will also be small.
- The medium on either side of the lens needs to be the same.

On violation of any of the limitation then the refraction at the curved surface formula is used for both the surfaces.

### Helpful Sign Conventions

- For a Convex (converging) Lens

R1 = positive

R2 = negative

f = positive

- For a Concave (diverging) Lens

R1 = negative

R2 = positive

f = negative

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