Refraction at Spherical Surfaces and Thin Lenses notes: ISC Class 12 Physics

1. Refraction at a single spherical surface

New Cartesian convention (light from left to right): distances from the pole P; along the light positive; real object u < 0; R positive when the centre of curvature C lies on the side the light goes to. Paraxial rays only.
Derivation (convex surface facing the rarer medium n₁, real image in the denser medium n₂): a ray from O meets the surface at A and bends towards the normal CA, crossing the axis at I. With α, β, γ the angles at O, I and C: i = α + γ (triangle OAC) and γ = r + β (triangle AIC). Small angles: n₁i = n₂r and α ≈ AD/OP, β ≈ AD/PI, γ ≈ AD/PC with D at P. Putting OP = −u, PI = v, PC = R gives $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ . With signs, the same formula is used for other cases.
Focal lengths of a surface: $f_2 = \dfrac{n_2 R}{n_2 - n_1}$ (parallel light from the first medium), $f_1 = -\dfrac{n_1 R}{n_2 - n_1}$ ; $f_2/f_1 = -n_2/n_1$ .
Magnification: $m = \dfrac{n_1 v}{n_2 u}$ . Flat surface (R → ∞): $n_2/v = n_1/u$ , the real and apparent depth relation. A point at the centre of curvature is seen at its true place (its rays meet the surface normally).

Formulas

Refraction at a spherical surface
$\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$
Magnification of a surface
$m = \dfrac{n_1 v}{n_2 u}$

Key terms

Paraxial rays: Rays close to the principal axis, making small angles with it, so that sin θ ≈ tan θ ≈ θ.
Radius of curvature (sign): Positive when the centre of curvature lies on the side to which the refracted light travels.

2. Lens maker’s formula, thin lens formula and lenses in liquids

Thin lens: two surfaces, thickness negligible; distances from the optical centre O. Light from the left: biconvex R₁ > 0, R₂ < 0; biconcave R₁ < 0, R₂ > 0; a plane face R = ∞.
Derivation (biconvex lens in air): surface 1 alone: $\dfrac{n}{v_1} - \dfrac{1}{u} = \dfrac{n - 1}{R_1}$ . Its image I₁ is the object for surface 2: $\dfrac{1}{v} - \dfrac{n}{v_1} = \dfrac{1 - n}{R_2}$ . Adding: $\dfrac{1}{v} - \dfrac{1}{u} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ . For u → −∞, v = f: lens maker’s formula $\dfrac{1}{f} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ , hence the thin lens formula $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ .
Special shapes: equiconvex f = R/[2(n − 1)]; plano-convex f = R/(n − 1); biconcave f negative. Reversing a lens does not change f.
In a medium of index n_m: replace n by n/n_m: $\dfrac{1}{f} = \left(\dfrac{n}{n_m} - 1\right)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$ , so $\dfrac{f_{\text{liquid}}}{f_{\text{air}}} = \dfrac{n - 1}{n/n_m - 1}$ . n_m < n: weaker, same type; n_m = n: no focusing; n_m > n: the lens changes type (an air bubble shaped like a convex lens in glass diverges light).

lens of n = 1.6, f = 20 cm in air

f(liquid) = f(air) × (n − 1)/(n/n_l − 1).

Formulas

Lens maker’s formula
$\dfrac{1}{f} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$
Thin lens formula
$\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$

Key terms

Optical centre: The point at the centre of a thin lens through which rays pass undeviated.
Lens maker’s formula: 1/f = (n − 1)(1/R₁ − 1/R₂), linking the focal length of a thin lens to its index and radii.

3. Magnification and power

Magnification: $m = \dfrac{h'}{h} = \dfrac{v}{u}$ (from the undeviated ray through O). m negative: real, inverted; m positive: virtual, erect.
Convex lens: real, inverted images for objects beyond F₁ (at 2F₁ → at 2F₂, same size); virtual, erect, enlarged inside F₁. Concave lens: always virtual, erect, diminished, between F₁ and O.
Power: $P = \dfrac{1}{f}$ with f in metres; unit dioptre (D) = m⁻¹. Converging +, diverging −. A shorter focal length means a larger power.
Graph: 1/v = 1/u + P is a straight line of slope 1, meeting both axes at a distance P from the origin.
Displacement method: for fixed object–screen distance D, two lens positions a distance x apart give sharp images; $f = \dfrac{D^2 - x^2}{4D}$ and m₁m₂ = 1.

convex lens, P = +6.0 D

Thin lens formula as a straight line: 1/v = 1/u + P.

Formulas

Magnification of a lens
$m = \dfrac{h'}{h} = \dfrac{v}{u}$
Power (f in metres)
$P = \dfrac{1}{f}$

Key terms

Power of a lens: The reciprocal of its focal length in metres; measured in dioptres, positive for converging lenses.
Dioptre: The unit of lens power: the power of a lens of focal length one metre (1 D = 1 m⁻¹).

4. Thin lenses in contact

Derivation: lens 1 alone: $\dfrac{1}{v_1} - \dfrac{1}{u} = \dfrac{1}{f_1}$ . I₁ is the object for lens 2 at the same distance (lenses touching): $\dfrac{1}{v} - \dfrac{1}{v_1} = \dfrac{1}{f_2}$ . Adding: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f_1} + \dfrac{1}{f_2}$ , so the pair acts as one lens with $\dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2}$ and $P = P_1 + P_2$ .
Magnification of the pair: m = m₁m₂. Equal and opposite powers cancel (P = 0, like a parallel-sided plate).
Lens on a plane mirror with a liquid: a liquid layer between a convex lens and a plane mirror forms a plano-concave liquid lens in contact with the glass lens; the pin–image coincidence distance is then the focal length F of the pair, giving the liquid lens from 1/f_liquid = 1/F − 1/f_glass.

Formulas

Lenses in contact
$\dfrac{1}{F} = \dfrac{1}{f_1} + \dfrac{1}{f_2}$
Powers add
$P = P_1 + P_2$

Key terms

Equivalent focal length: The focal length F of the single lens that has the same effect as the combination: 1/F = 1/f₁ + 1/f₂.

5. Combinations of separated lenses, and a lens with a mirror

Separated lenses (distance d): find v₁ for lens 1; the object for lens 2 is at $u_2 = v_1 - d$ (measured from lens 2); if I₁ lies beyond lens 2, u₂ > 0 (virtual object: the rays reach lens 2 still converging). Total magnification m = m₁m₂.
Lens with a mirror: light passes the lens, is reflected, and passes the lens again. Treat each step in turn: lens image → object for the mirror (a plane mirror forms its image as far in front as the object is behind) → object for the lens on the return pass, measuring distances along the returning light. m = m₁ × m(mirror) × m₃.
Auto-collimation: a pin at the focus of a convex lens backed by a plane mirror coincides with its own inverted image, whatever the lens–mirror distance: a quick way to measure f.
Image back on the object: if the lens’s image falls at the centre of curvature of a concave or convex mirror placed behind it, the rays retrace their path and the final image forms on the object (inverted). Silvered lenses are not in the ISC syllabus.

Formulas

Object distance for the second lens
$u_2 = v_1 - d$
Total magnification
$m = m_1 m_2$

Key terms

Virtual object: A point towards which rays are converging when they meet a lens or mirror; it lies beyond the element and has u > 0.
Auto-collimation: Finding the focal length of a lens by placing a plane mirror behind it and making an object coincide with its own image.

Exam tips

Signs first: n₁ is the medium the light comes from; R positive when C is on the outgoing side; real object u < 0.
Lens: m = v/u (mirror: m = −v/u); P = 1/f with f in metres.
In a liquid use n/n_m; a liquid denser than the glass reverses the lens.
Combinations: one image at a time; u₂ = v₁ − d; multiply magnifications; on a return pass measure along the returning light.

1. Refraction at a single spherical surface

Formulas

Key terms

2. Lens maker’s formula, thin lens formula and lenses in liquids

Formulas

Key terms

3. Magnification and power

Formulas

Key terms

4. Thin lenses in contact

Formulas

Key terms

5. Combinations of separated lenses, and a lens with a mirror

Formulas

Key terms

Exam tips

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