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Ray Optics and Optical Instruments — Physics STD 12 Science — Question

Bihar BoardEnglish MediumSTD 12 SciencePhysicsRay Optics and Optical Instruments5 Marks
Question
Write the formula for refraction of light at spherical surface (concave or convex). With the help of this formula derive the lens formula : $\frac{1}{f}=(n-1)(\frac{1}{R_{1}}-\frac{1}{R_{2}})$, where symbols used have their usual meanings.
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Answer

Formula for refraction at spherical surface (concave or convex) is as follows :
$\frac{n}{v}-\frac{1}{u}=\left(\frac{n-1}{ R }\right)$
where $n$ = refractive index of the medium behind the surface with respect to the medium infront of the surface, R = radius of the curvature of spherical surface, $u$ = distance of object from the surface and $v$ = distance of the image from the surface.
Derivation of the formula : $\frac{1}{f}=(n-1)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$
In the ray diagram given below a convex lens 'L' made of absolute refractive index $n_2$ is placed in rarer medium (say air) of absolute refractive index $n_1$ i.e. $n_1 < n_2$.
Image
In this figure P1 , P2 are the poles and C1 , C2 are the centres of curvature of the lens. Hence P1C1 = R1 and P2C2 = R2 are the radii of curvatures of first and second surface of the lens respectively marked as (1) and (2) and P1P2 = t is the thickness of the lens. Let C be the optical centre of the lens.
Suppose O is a point object placed on the principal axis of the lens at a distance u from the pole P1 of surface (1) of the lens. A ray OA is incident on this surface. At A it is refracted from rarer medium into denser medium bending towards normal C1AN1 at A along AB inside the lens. If the surface (2) were absent the ray AB would have met the principal axis at $I^{\prime}$ at a distance $v^{\prime}$ from the pole P1. Hence $I^{\prime}$  can be treated as the real image formed by surface (1). On the basis of formula for refraction at single refracting surface for refraction at surface (1) from rarer medium of refractive index n1 to denser medium of refractive index n2 we have
$\frac{n_2}{v^{\prime}}-\frac{n_1}{u}=\frac{n_2-n_1}{R_1}$ ...(1)
The ray AB travelling in medium n2 is actually incident on the second surface of the lens and is refracted into the rarer medium n1, bending away from the normal C2BN2 at the point B. The emergent ray meets the principal axis at I which is the final, real image of O formed by the lens.
For refraction at the second surface of the lens, $I^{\prime}$ acts as virtual object and I is the real image of $I^{\prime}$ formed by this surface. Let v be the distance of I from the pole P2 of the second surface. The distance of $I^{\prime}$ (object for the second surface) from the pole P2 of the second surface is $\left(v^{\prime}-t\right)$. Then, for refraction at the surface (2) from denser medium $n_2$ to rarer medium $n_1$, we have
$\frac{n_1}{v}-\frac{n_2}{v^{\prime}-t}=\frac{n_1-n_2}{ R _2}$
But for thin lens thickness t << $v^{\prime}$ , hence this can be neglected, so we have
$\frac{n_1}{v}-\frac{n_2}{v^{\prime}}=\left(\frac{n_1-n_2}{ R _2}\right)$
or $\frac{n_1}{v}-\frac{n_2}{v^{\prime}}=-\left(\frac{n_2-n_1}{R_2}\right)$ ...(2)
On adding equation (1) and (2) we get
$\frac{n_1}{v}-\frac{n_1}{u}=\left(n_2-n_1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$
Dividing both sides by n1 , we get
$\frac{1}{v}-\frac{1}{u}=\left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$
But n1/n2= n the refractive index of the material of the lens with respect to the surrounding medium,
$\therefore \quad \frac{1}{v}-\frac{1}{u}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$ ... (3)
But when the object is at infinity the image will be formed at the second focus of the lens i.e., when $u =\infty$, $v$ $=f$ the focal length of the lens. Putting these values of u and v in equation (3), we get
$\frac{1}{f}-\frac{1}{\infty}=(n-1)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$
$\Rightarrow \quad \frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$ ...(4)
Equation (4) represents the lens maker's formula.
It has been derived for a convex lens forming a real image, but it is equally applicable to a convex lens forming a virtual image and to a concave lens which forms only virtual image.

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