Q.An object approaches a convergent lens from the left of the lens with a uniform speed 5 m/s and stops at the focus. The image

The Lens Maker's formula, $\dfrac{1}{f}=(\mu-1)\left(\dfrac{1}{R_1}-\dfrac{1}{R_2}\right)$, is derived by applying single-surface refraction twice, once at each face of the lens. (OR alternative: Huygens' Principle constructs wavefronts from secondary wavelets, and can be used to derive the laws of reflection geometrically.)

Part 1: Lens Maker's Formula

What it is. The Lens Maker's formula relates a lens's focal length $f$ to the refractive index $\mu$ of its material (relative to the surrounding medium) and the radii of curvature $R_1$, $R_2$ of its two spherical surfaces — it tells a lens manufacturer what curvatures are needed to grind a lens of a desired focal length.

Derivation for a thin convex lens. Consider a thin lens with two refracting surfaces of radii $R_1$ (first surface, light hits this first) and $R_2$ (second surface), made of material of refractive index $\mu$, surrounded by air (index 1). Let a point object O on the principal axis form an image after refraction at each surface in turn.

Step 1 — Refraction at the first surface (radius $R_1$), treating it alone (ignoring the second surface for now), forming a virtual intermediate image $I_1$ at distance $v_1$:

$\frac{\mu}{v_1} - \frac{1}{u} = \frac{\mu-1}{R_1}$

Step 2 — Refraction at the second surface (radius $R_2$): the image $I_1$ from step 1 now acts as the object for this second refraction (light going from the denser lens medium $\mu$ back into air, index 1), forming the final image $I$ at distance $v$:

$\frac{1}{v} - \frac{\mu}{v_1} = \frac{1-\mu}{R_2}$

Step 3 — Add the two equations (the $\mu/v_1$ terms cancel):

$\frac{1}{v} - \frac{1}{u} = (\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Step 4 — Apply the lens definition. When the object is at infinity ($u\to\infty$), the image forms at the focus ($v=f$), so $\frac{1}{v}-\frac{1}{u} \to \frac{1}{f}$. Substituting:

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

This is the Lens Maker's formula, and it also leads to the general thin lens formula $\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$.

OR: Huygens' Principle and the Laws of Reflection

Huygens' Principle. Every point on a given wavefront (a surface of constant phase) acts as a source of new secondary wavelets, which spread out in all directions with the speed of the wave in that medium. The new wavefront at any later time is given by the forward "envelope" (common tangent surface) of all these secondary wavelets.

Deriving the laws of reflection. Consider a plane wavefront AB incident on a reflecting surface MN at angle of incidence $i$, with A striking the surface first while B is still travelling.

• By the time the wavelet from B reaches the surface at C (after time $\tau$), the wavelet from A (already at the surface) has spread out as a hemisphere of radius $v\tau = AD$ (since it reflects back into the same medium, same speed $v$).

• The new reflected wavefront is the common tangent DC from this envelope.

• Comparing right triangles $ABC$ and $ADC$ (sharing hypotenuse AC): $AD = BC = v\tau$ (equal radii, both wavelets travelled for the same time in the same medium), and both are right triangles, so they are congruent, giving $\angle BAC = \angle DCA$.

• Since $\angle BAC$ is the angle of incidence $i$ (angle between incident wavefront and surface) and $\angle DCA$ is the angle of reflection $r$ (angle between reflected wavefront and surface), this congruence directly gives:

  $i = r \quad \text{(angle of incidence = angle of reflection)}$

• Also, the incident ray, the reflected ray, and the normal at the point of incidence can be shown to lie in the same plane, completing the laws of reflection.

The lens maker's formula, 1/f = (n21-1)(1/R1 - 1/R2), is derived by applying refraction at each of the two spherical surfaces of a thin lens in turn. (OR: Fringe width beta = lambdaD/d is derived from the path-difference condition for constructive/destructive interference in Young's double-slit experiment.)

Derivation of Lens Maker's Formula (for a thin double convex lens):

Assumptions: (i) The lens is thin, so the two refracting surfaces are close enough that the lateral displacement of the ray inside the lens can be neglected. (ii) Only paraxial rays (making small angles with the principal axis) are considered. (iii) The medium on both sides of the lens is the same (say, refractive index n1), and the lens material has refractive index n2.

Sign convention: All distances are measured from the pole/optical centre of the surface; distances measured in the direction of the incident light are taken positive, against it negative. For a double convex lens, the first surface (facing the incident light) is convex towards the object, so its radius R1 is positive; the second surface curves the other way, so its radius R2 is negative.

Consider a point object O on the principal axis. Let the first surface (radius R1) refract the light from the object; treating this surface alone, the image I1 formed (a virtual, intermediate image) obeys the single-surface refraction formula:

n2/v1 - n1/u = (n2 - n1)/R1 ... (i)

where u is the object distance and v1 is the image distance for the first surface alone.

This intermediate image I1 now acts as a virtual object for the second surface (radius R2), which refracts the ray back into the surrounding medium n1, forming the final real image I at distance v:

n1/v - n2/v1 = (n1 - n2)/R2 ... (ii)

Adding equations (i) and (ii), the n2/v1 and -n2/v1 terms cancel:

n1/v - n1/u = (n2-n1)/R1 + (n1-n2)/R2 = (n2-n1)*(1/R1 - 1/R2).

Dividing throughout by n1:

1/v - 1/u = (n2/n1 - 1)(1/R1 - 1/R2) = (n21 - 1)(1/R1 - 1/R2), where n21 = n2/n1 is the refractive index of the lens material relative to the surrounding medium.

Now, if the object is placed at infinity (u tends to infinity), the rays refracted by the lens converge (for a convex lens) to the principal focus, so v = f (the focal length). Putting u = infinity, 1/u = 0:

1/f = (n21 - 1)*(1/R1 - 1/R2).

This is the Lens Maker's Formula. It relates the focal length f of a thin lens to the refractive index of its material (relative to the surrounding medium) and the radii of curvature of its two surfaces, and is used by lens manufacturers to design a lens of a required focal length. For a double convex lens (R1 positive, R2 negative), (1/R1 - 1/R2) is positive, so f comes out positive - confirming it is a converging lens.

The lens-maker's formula relates a thin lens's focal length to its refractive index and the radii of curvature of its two surfaces, derived by applying single-surface refraction twice.

Sign convention (Cartesian, as used in NCERT): All distances are measured from the optical centre of the lens. Distances measured in the direction of the incident light are taken as positive; distances measured against the direction of incident light are taken as negative. Heights measured upward from the principal axis are positive, downward are negative. For a convex lens, if the centre of curvature of a surface lies on the outgoing-light side, its radius $R$ is positive; if on the incoming-light side, $R$ is negative.

Derivation of lens-maker's formula:

Consider a thin convex lens of refractive index $n_2$ placed in a medium of refractive index $n_1$, with surfaces of radii $R_1$ and $R_2$. Let an object be at O on the principal axis.

Refraction at the first surface (radius $R_1$) forms an image at $I_1$ (treating the second surface as absent), using the single spherical refracting surface formula:

$\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2-n_1}{R_1}$

Refraction at the second surface (radius $R_2$): the image $I_1$ from the first surface now acts as a virtual object for the second surface, forming the final image at $I$ (at distance $v$):

$\frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1-n_2}{R_2}$

Adding these two equations (the $n_2/v_1$ terms cancel):

$\frac{n_1}{v} - \frac{n_1}{u} = (n_2-n_1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Dividing throughout by $n_1$:

$\frac{1}{v}-\frac{1}{u} = \left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right) = (n_{21}-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

When the object is at infinity ($u\to\infty$), the image forms at the focus, $v=f$, giving the general lens formula $\dfrac1v-\dfrac1u=\dfrac1f$, so:

$\boxed{\frac{1}{f} = (n_{21}-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)}$

where $n_{21}=n_2/n_1$ is the refractive index of the lens material relative to the surrounding medium.

OR — Huygens' principle and laws of refraction:

Huygens' wave principle: Every point on a given wavefront (locus of points vibrating in phase) acts as a source of new secondary wavelets, which spread out in all directions with the speed of the wave in that medium. The new (secondary) wavefront at any later instant is the surface tangent (envelope) to all these secondary wavelets.

Derivation of Snell's law using Huygens' construction:

Consider a plane wavefront AB incident on a plane interface XY separating medium 1 (speed $v_1$) from medium 2 (speed $v_2$), making angle of incidence $i$ with the normal. Let the time taken for the wavelet from B to reach the interface at C be $\tau$, so $BC = v_1\tau$.

During this same time $\tau$, the secondary wavelet from A (which reached the interface first) spreads into medium 2 with radius $AD = v_2\tau$. The new refracted wavefront is the tangent CD from C to this wavelet.

From the right triangle ABC: $\sin i = \dfrac{BC}{AC} = \dfrac{v_1\tau}{AC}$.

From the right triangle ACD (with $r$ = angle of refraction): $\sin r = \dfrac{AD}{AC} = \dfrac{v_2\tau}{AC}$.

Dividing these:

$\frac{\sin i}{\sin r} = \frac{v_1}{v_2}$

Since refractive index $n = c/v$, we have $v_1 = c/n_1$ and $v_2 = c/n_2$, so $\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}$, giving

$n_1\sin i = n_2\sin r$

which is Snell's law of refraction. (The incident ray, refracted ray, and normal are all seen to lie in the plane of incidence from this construction, verifying the first law of refraction too.)