# The Lens Maker's Formula

Lens manufacturers use the lens maker’s formula to manufacture lenses of the desired focal length. Lenses of different focal length are used for different optical instruments. The focal length of a lens depends on the refractive index of the lens and the radii of curvature. The lens maker’s equation is another formula used for lenses that give 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 are 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.

## Limitations of Lens Maker’s Formula

1. The lens needs to be thin. This is because the separation between the two refracting surfaces will also be small.
2. 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.

• For a Convex (converging) Lens

R1 = positive

R2 = negative

f = positive

• For a Concave (diverging) Lens

R1 = negative

R2 = positive

f = negative

If the focal length is positive then the lens is said to be converging and if the focal length is negative then it is said to be diverging. From this, we can come to a conclusion that a convex lens doesn’t necessarily have to be converging and concave diverging.

Stay tuned with Byju’s to learn about concave and convex lens and much more.

#### Practise This Question

The heat produced in a wire carrying an eletric current depends on the current, the resistance and the

time. Assuming that the dependence is of the product of powers type, guess an equation between these

quantities using dimensional analysis. The dimensional formula of resistance is ML2I2T3and heat is a

form of energy.