Derivation Of Lens Maker Formula

Lens Maker Formula

For various optical equipment, lenses of varying focal lengths are utilized. The focal length of a lens is determined by the refractive index of the lens's material and the curvature radii of the two surfaces. The lens maker formula is derived here to help applicants better comprehend the subject. The lens maker formula is often used by lens manufacturers to create lenses with the appropriate focal length.

For spherical lenses, the lens equation or lens formula is an equation that links the focal length, image distance, and object distance.
Lens Formula $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$ is how it's written. where. $v$ is the image's distance from the lens, $u$ is the object distance and $f$ is the focal length.

If the relationship between a lens' focal length, the refractive index of its material, and the radii of curvature of its two surfaces is known as the lens maker's formula. Lens manufacturers utilize it to build lenses with a specific power from glass with a specific refractive index.

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

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Derivation of Lens Maker’s Formula

The lens maker formula is derived using the assumptions listed below

Consider the thin lens in the picture above, which has two refracting surfaces with curvature radii R₁ and R₂, respectively. Assume that the surrounding medium and the lens material have refractive indices of n₁ and n₂, respectively. The whole derivation of the lens maker formula is provided further below. We can say that, using the formula for refraction at a single spherical surface,

For the first surface,
$
\frac{n_2}{v_1}-\frac{r_1}{u}=\frac{n_2-r_1}{R_1} \ldots
$(1)
For the second surface,

$
\frac{n_1}{v}-\frac{n_2}{v_1}=\frac{n_1-n_2}{R_2} \ldots
$(2)
Now adding equation (1) and (2),

$
n 1 / v-n 1 / u=\left(n_2-n_1\right)\left[1 / R_1-1 / R_2\right]
$

on simplifying we get,

$
1 / v-1 / u=\left(n_2 / n_1-1\right)\left[1 / R_1-1 / R_2\right]
$
When $u=\infty$ and $v=f$

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

But also,
$
\frac{1}{v}-\frac{1}{u}=\frac{1}{f}
$
Therefore, we can say that,

$
\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$
Where $\mu$ is the material's refractive index.

Where μ is the material's refractive index.

This is the derivation of the lens maker formula. Examine the constraints of the lens maker's formula to have a better understanding of the lens maker's formula derivation.

Image Formation With a Thin Lens: Characteristics

It's not enough to know the thin lens formula for convex lenses. The characteristics of a ray of light going through converging and diverging lenses must be understood.

1. On the opposite side, parallel rays going through converging lenses will intersect at point f.

2. Parallel rays seem to emerge from point f in front of diverging lenses.

3. The direction of light rays travelling through the centre of converging or diverging lenses does not change.

4. Light rays that enter a converging lens through its focal point always exit parallel to the lens's axis.

5. On the other side of a diverging lens, a light ray going towards the focal point will also emerge parallel to its axis.

A concave or divergent lens has a negative focal length. When the picture is generated on the side where the object is positioned, the image distance is also negative. The image is virtual in this case. A converging or convex lens, on the other hand, has a positive focal length.

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Limitations of the Lens Maker’s Formula

• The lens should not be thick so that the space between the 2 refracting surfaces can be small.
• The medium used on both sides of the lens should always be the same.