(a) Deduce the relation μ₂/v − μ₁/u = (μ₂−μ₁)/R, when refraction takes place from rarer to denser medium (μ₂>μ

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Q.(a) Deduce the relation μ₂/v − μ₁/u = (μ₂−μ₁)/R, when refraction takes place from rarer to denser medium (μ₂>μ₁) in case of a convex surface. [3]

(b) Using the above formula, deduce the lens maker formula 1/f = (μ−1)(1/R₁ − 1/R₂). [2] OR

(a) What is fringe width? [1]

(b) Prove that the width for bright fringes is the same as for dark fringes in the interference pattern in Young's double slit experiment. [2]

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Concept understanding — Refraction at a Spherical Surface

Refraction at a Spherical Surface

Imagine you are looking at a fish in a pond. The fish appears closer to the surface than it actually is. That is refraction — light bends when it moves from water into air. Now imagine the boundary between the two media is not flat, but curved like the surface of a lens or a glass marble. That is a spherical refracting surface.

The Intuition

When light hits a flat surface (like a glass slab), it bends once and travels straight. But when the surface is curved, something more interesting happens. Each ray of light strikes the curve at a slightly different angle, so each ray bends by a different amount. The result is that all rays leaving one point can be made to converge to (or diverge from) another point — forming an image.

Think of a spherical surface as a tiny piece of a sphere. The centre of that sphere is called the centre of curvature $C$ , and the distance from the surface to $C$ is the radius of curvature $R$ . The line joining the centre of curvature to the centre of the surface is the principal axis.

The Sign Convention

Before we write the formula, we need a consistent way to measure distances. The standard convention (Cartesian sign convention) is:

Distances measured against the direction of incident light are negative.
Distances measured along the direction of incident light are positive.
The pole $P$ (the vertex of the spherical surface) is the origin.

So for a convex surface (bulging toward the incident light), $R$ is positive. For a concave surface (curving away), $R$ is negative. Object distance $u$ is always negative (object is on the incident side). Image distance $v$ can be positive or negative depending on where the image forms.

The Derivation in One Paragraph

Consider a point object $O$ on the principal axis. A ray from $O$ strikes the spherical surface at point $A$ and bends according to Snell's law: $n_1 \sin i = n_2 \sin r$ . For small angles (paraxial approximation), $\sin\theta \approx \theta$ , so $n_1 i = n_2 r$ . Using geometry, $i = \alpha + \gamma$ and $r = \gamma - \beta$ , where $\alpha$ , $\beta$ , $\gamma$ are the angles the ray makes with the principal axis at $O$ , $I$ , and $C$ respectively. Substituting and using $\alpha \approx \tan\alpha = \frac{AP}{PO}$ , $\beta \approx \frac{AP}{PI}$ , $\gamma \approx \frac{AP}{PC}$ , and cancelling $AP$ , you get the relation.

\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}

Here:

$n_1$ = refractive index of the medium where the object lies
$n_2$ = refractive index of the medium where the image forms
$u$ = object distance from pole (negative)
$v$ = image distance from pole (positive if real image on the opposite side)
$R$ = radius of curvature (positive if centre of curvature is on the image side)

What This Formula Tells You

The formula is a single equation that connects four things: where the object is, where the image forms, how curved the surface is, and what the two media are. If you know any three, you can find the fourth.

For a convex surface (say, light going from air into glass through a convex surface), $R > 0$ and $n_2 > n_1$ , so the right side is positive. This means $v$ is positive — the image forms on the other side. The surface converges light.

For a concave surface (light going from air into glass through a concave surface), $R < 0$ , so the right side is negative. The surface diverges light. …

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Applying Snell's law in the small-angle (paraxial) approximation at a single spherical refracting surface gives the refraction formula; applying it twice, once for each surface of a thin lens, and adding the results gives the lens maker's formula.

Refraction at a single spherical convex surface (rarer to denser, $\mu_2 > \mu_1$ ): Consider a spherical refracting surface separating a rarer medium (refractive index $\mu_1$ ) from a denser medium ( $\mu_2$ ), with centre of curvature C, pole P, and radius of curvature $R = PC$ . Let a point object O lie on the principal axis in the rarer medium, and let a paraxial ray OA refract at A (close to P) and converge to (or appear to diverge from) the image point I in the denser medium. Let $\angle NOA = \alpha$ (angle of incidence measured from the axis), $\angle NIA=\beta$ , $\angle NCA=\gamma$ , where N is the point A projected onto the axis (foot of the perpendicular, at paraxial approximation coinciding with P). For small (paraxial) angles, $\tan\theta \approx \theta \approx \sin\theta$ , so from triangle geometry: Exterior angle relations give: angle of incidence $i = \alpha + \gamma$ , and angle of refraction $r = \gamma - \beta$ . By Snell's law (paraxial form): $\mu_1 i = \mu_2 r$ (since $\sin\theta\approx\theta$ ): $\mu_1(\alpha+\gamma) = \mu_2(\gamma-\beta)$ Using the paraxial approximations $\alpha \approx \dfrac{h}{-u}$ [i.e. $h/OP$ ], wait — using standard sign convention (distances measured from pole P, along the direction of incident light positive): $\alpha \approx h/(-u)$ , $\beta \approx h/v$ , $\gamma \approx h/R$ , where $h$ is the height of A above the axis, and $u, v, R$ are the object, image, and radius-of-curvature distances (with sign). Substituting and simplifying (dividing through by $h$ ): $\mu_1\left(\dfrac{1}{-u}+\dfrac{1}{R}\right) = \mu_2\left(\dfrac{1}{R}-\dfrac{1}{v}\right)$ Rearranging terms: $\dfrac{\mu_2}{v} - \dfrac{\mu_1}{u} = \dfrac{\mu_2-\mu_1}{R}$ This is the required refraction formula for a single spherical surface, valid for paraxial rays, when light goes from a rarer medium ( $\mu_1$ ) to a denser medium ( $\mu_2$ ) at a convex surface.
Lens maker's formula: A thin lens has TWO refracting surfaces. Consider a thin lens of refractive index $\mu$ (relative to the surrounding medium, taken as air, $\mu_1=1$ ) with radii of curvature $R_1$ (first surface) and $R_2$ (second surface). Let an object O produce, after refraction at the FIRST surface alone, a virtual image $I_1$ at distance $v_1$ (as if the lens were only the first surface, with the second medium being the lens material of index $\mu$ ): …

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Q.(a) Deduce the relation μ₂/v − μ₁/u = (μ₂−μ₁)/R, when refraction takes place from rarer to denser medium (μ₂>μ₁) in case of a convex surface. [3]

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